Formality properties of finitely generated groups and Lie algebras

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FORMALITY PROPERTIES OF FINITELY GENERATED GROUPS ANDLIE ALGEBRAS

ALEXANDER I. SUCIU1 AND HE WANG

Abstract. We explore the graded and filtered formality properties of afinitely-generated groupby studying the various Lie algebras attached to such a group, including the associated graded Liealgebra, the holonomy Lie algebra, and the Malcev Lie algebra. We explain how these notionsbehave with respect to split injections, coproducts, and direct products, and how they are inheritedby solvable and nilpotent quotients.

For a finitely-presented group, we give an explicit formula for the cup product in low de-grees, and an algorithm for computing the holonomy Lie algebra, using a Magnus expansionmethod. We also give a presentation for the Chen Lie algebra of a filtered-formal group, and dis-cuss various approaches to computing the ranks of the gradedobjects under consideration. Weillustrate our approach with examples drawn from a variety of group-theoretic and topologicalcontexts, such as 1-relator groups, finitely generated torsion-free nilpotent groups, link groups,and fundamental groups of Seifert fibered manifolds.

Contents

1. Introduction 22. Filtered and graded Lie algebras 63. Graded algebras and Koszul duality 104. Groups, Lie algebras, and graded formality 135. Malcev Lie algebras and filtered formality 176. Derived series and Lie algebras 227. Torsion-free nilpotent groups 258. Magnus expansions for finitely generated groups 289. Group presentations and (co)homology 3210. A presentation for the holonomy Lie algebra 3611. Mildness, one-relator groups, and link groups 3812. Seifert fibered manifolds 42References 46

2010Mathematics Subject Classification.Primary 20F40. Secondary 16S37, 17B70, 20F14, 20J05, 55P62, 57M05.Key words and phrases.Lower central series, derived series, Malcev Lie algebra, holonomy Lie algebra, Chen

Lie algebra, Koszul algebra, Magnus expansion, cohomologyring, 1-formality, graded formality, filtered formality,nilpotent group, Seifert manifold.

1Partially supported by NSF grant DMS–1010298 and NSA grant H98230-13-1-0225.

1

http://arxiv.org/abs/1504.08294v1

2 ALEXANDER I. SUCIU AND HE WANG

1. Introduction

The main focus of this paper is on the formality properties offinitely generated groups, asreflected in the structure of the various Lie algebras attached to such groups.

1.1. From groups to Lie algebras. Throughout, we will letG be a finitely generated group,and we will fix a coefficient fieldk of characteristic 0. Our main focus will be on severalk-Liealgebras attached to such a group, and the way they all connect to each other.

By far the best known of these Lie algebras is theassociated graded Lie algebra, gr(G; k),introduced by P. Hall, W. Magnus, and E. Witt in the 1930s, cf.[43]. This is a finitely generatedgraded Lie algebra, whose graded pieces are the successive quotients of the lower central seriesof G (tensored withk), and whose Lie bracket is induced from the group commutator. Thequintessential example is the free Lie algebralie(kn), which is the associated graded Lie algebraof the free group onn generators,Fn.

Closely related is theholonomy Lie algebra, h(G; k), introduced by K.-T. Chen in [13], andfurther studied in [29, 45, 53]. This is a quadratic Lie algebra, obtained as the quotient of thefree Lie algebra onH1(G; k) by the ideal generated by the image of the dual of the cup prod-uct map in degree 1. The holonomy Lie algebra comes equipped with a natural epimorphismΦG : h(G; k) ։ gr(G; k), and thus can be viewed as the quadratic approximation to the associ-ated graded Lie algebra ofG.

The most intricate of these Lie algebras (yet, in many ways, the most important) is theMalcevLie algebra, m(G; k). As shown by A. Malcev in [44], every finitely-generated, torsion-freenilpotent groupN is the fundamental group of a nilmanifold, whose correspondingk-Lie algebrais m(N; k). Taking now the nilpotent quotients ofG, we may definem(G; k) as the inverselimit of the resulting tower of nilpotent Lie algebras,m(G/ΓkG, k). By construction, this is acomplete, filtered Lie algebra. In two seminal papers, [61, 62], D. Quillen showed thatm(G; k)is the set of all primitive elements inkG (the completion of the group algebra ofG with respectto the filtration by powers of the augmentation ideal), and that the associated graded Lie algebraof m(G; k) is isomorphic to gr(G, k).

1.2. Formality notions. In his foundational paper on rational homotopy theory [70], D. Sulli-van associated to each connected CW-complexX a ‘minimal model’,M (X), that can be viewedas an algebraic approximation toX. In particular, ifX has finite 1-skeleton, then the Lie algebradual to the first stage of the minimal model is isomorphic to the Malcev Lie algebram(G,Q)associated to the fundamental groupG = π1(X).

The spaceX is said to beformal if the commutative, graded differential algebraM (X) isquasi-isomorphic to the cohomology ringH∗(X,Q), endowed with zero differential. If the cdgamorphismM (X)→ H∗(X,Q) merely induces isomorphisms in cohomology up to degreeq anda monomorphism in degreeq+ 1, thenX is calledq-formal.

The study of the various Lie algebras attached to the fundamental group of a space providesa fruitful way to look at the formality problem. Indeed, the spaceX is 1-formal if and only if thegroupG = π1(X) is 1-formal (overQ), that is, the Malcev Lie algebram(G;Q) is isomorphicto the rational holonomy Lie algebra ofG, completed with respect to bracket length. Conse-quently, the 1-formality ofX (or G) is equivalent to the quadraticity ofm(G;Q), which, in turn,is equivalent to the quadraticity ofm(G; k), for any field extensionQ ⊂ k.

We find it useful to separate the 1-formality property of a finitely generated groupG intotwo complementary properties: graded formality and filtered formality. More precisely, we saythatG is graded-formal(overk) if the associated graded Lie algebra gr(G; k) is isomorphic, as

FORMALITY PROPERTIES OF FINITELY GENERATED GROUPS AND LIE ALGEBRAS 3

a graded Lie algebra, to the holonomy Lie algebrah(G; k). Likewise, we say thatG is filtered-formal (overk) if the Malcev Lie algebram(G; k) is isomorphic, as a filtered Lie algebra, to thecompletion of its associated graded Lie algebra,gr(m(G; k)). As we show in Proposition5.6,the groupG is 1-formal if and only if it is both graded-formal and filtered-formal.

All four possible combinations of these formality properties occur:

(1) Examples of 1-formal groups include finitely generated free groups and free abeliangroups (more generally, right-angled Artin groups), groups with first Betti numberequal to 0 or 1, fundamental groups of compact Kahler manifolds, and fundamentalgroups of complements of complex algebraic hypersurfaces.

(2) There are many torsion-free, nilpotent groups (Example5.7, or, more generally,7.9)as well as link groups (Examples11.13and11.14) which are filtered-formal, but notgraded-formal.

(3) There are also finitely presented groups, such as those from Examples5.3 and11.5,which are graded-formal but not filtered-formal.

(4) Finally, there are groups which enjoy none of these formality properties. Indeed, ifG1 isone of the groups from (2) andG2 is one of the groups from (3), then TheoremB belowshows that the productG1×G2 and the free productG1 ∗G2 are neither graded-formal,nor filtered-formal.

1.3. Propagation of formality. After reviewing in Sections2 and3 some basic notions pertain-ing to filtered and graded Lie algebras, as well as the notionsof quadratic and Koszul algebras,we develop in Sections4 and 5 the basic theory of filtered and graded formality of finitelygenerated groups, with special emphasis on the way these notions behave with respect to splitinjections, coproducts, and direct products.

Our first main result is a combination of Theorem4.11and5.10, and can be stated as follows.

Theorem A. Let G be a finitely generated group, and let K≤ G be a subgroup. Suppose thereis a split monomorphismι : K → G. Then:

(1) If G is graded-formal, then K is also graded-formal.(2) If G is filtered-formal, then K is also filtered-formal.(3) If G is 1-formal, then K is also1-formal.

In particular, if a semi-direct productG1⋊G2 has one of the above formality properties, thenG2 also has that property; in general, though,G1 will not, as illustrated in Example4.13.

As shown in [17], both the product and the coproduct of two 1-formal groups is again 1-formal. Also, as shown in [57], the product and coproduct of two graded-formal groups isagain graded-formal. We sharpen these results in the next theorem, which is a combination ofPropositions4.15and5.12.

Theorem B. Let G1 and G2 be two finitely generated groups. The following conditions areequivalent.

(1) G1 and G2 are graded-formal (respectively, filtered-formal, or1-formal).(2) G1 ∗G2 is graded-formal (respectively, filtered-formal, or1-formal).(3) G1 ×G2 is graded-formal (respectively, filtered-formal, or1-formal).

Both TheoremA andB can be used to decide the formality properties of new groups fromthose of known groups. In general, though, even when bothG1 andG2 are 1-formal, we cannotconclude that an arbitrary semi-direct productG1 ⋊G2 is 1-formal (see Example11.15).


The various formality properties are not necessarily inherited by the quotient groups. How-ever, as we shall see in TheoremsC andD, respectively, filtered formality is passed on to thederived quotients and to the nilpotent quotients of a group.

1.4. Derived series and Lie algebras.In Section6, we investigate some of the relationshipsbetween the lower central series and derived series of a group, on one hand, and the derivedseries of the corresponding Lie algebras, on the other hand.

In [12], K.-T. Chen studied the lower central series quotients of the maximal metabelianquotient of a finitely generated free group, and computed their graded ranks. More generally,following [53], we may define theChen Lie algebrasof a groupG as the associated graded Liealgebras of its solvable quotients, gr(G/G(i); k). Our next theorem (which combines Theorem6.4and Corollary6.6) sharpens and extends the main result of [53].

Theorem C. Let G be a finitely generated group. For each i≥ 2, the quotient map G։ G/G(i)

induces a natural epimorphism of gradedk-Lie algebras,

Ψ(i)G : gr(G; k)/ gr(G; k)(i) // // gr(G/G(i); k) .

Moreover,

(1) If G is a filtered-formal group, then each solvable quotient G/G(i) is also filtered-formal,and the mapΨ(i)

G is an isomorphism.(2) If G is a 1-formal group, thenh(G; k)/h(G; k)(i)

� gr(G/G(i); k).

Given a finitely presented groupG, the solvable quotientsG/G(i) need not be finitely pre-sented. Thus, finding presentations for the Chen Lie algebragr(G/G(i)) can be an arduous task.Nevertheless, TheoremC provides a method for finding such presentations, under suitable for-mality assumptions. The theorem can also be used as an obstruction to 1-formality.

1.5. Nilpotent groups. Our techniques apply especially well to the class of finitelygener-ated, torsion-free nilpotent groups. Carlson and Toledo [9] studied the 1-formality properties ofsuch groups, while Plantiko [57] gave a sufficient conditions for such groups to be non-graded-formal. Recently, Cornulier [15] proved that the systolic growth of a finitely generated nilpotentgroupG is asymptotically equivalent to its growth if and only if theMalcev Lie algebram(G, k)is filtered-formal (or, ‘Carnot’).

We investigate in Section7 the filtered formality of nilpotent groups, and the way this prop-erty interacts with other properties of these groups. The next result combines Theorems7.2and7.5, as well as Proposition7.10.

Theorem D. Let G be a finitely generated group.

(1) If G is filtered-formal, then all the nilpotent quotients G/Γk(G) are also filtered-formal.(2) Suppose G is a torsion-free,2-step nilpotent group with torsion-free abelianization.

Then G is filtered-formal.(3) Suppose G is a torsion-free, filtered-formal, nilpotent group. Then the universal en-

veloping algebra U(gr(G; k)) is Koszul if and only if G is abelian.

In particular, eachn-step, free nilpotent groupF/ΓnF is filtered-formal. A classical exampleis the unipotent groupUn(Z), which is filtered-formal [35], but not graded-formal forn ≥ 3.


1.6. Presentations. In Sections8 to 10we analyze the presentations of the various Lie algebrasattached to a finitely presented groupG. Some of the motivation and techniques come from thework of J. Labute [30, 31] and D. Anick [1], who gave presentations for the associated gradedLie algebra gr(G; k), providedG has a ‘mild’ presentation.

Our main interest, though, is in finding presentations for the holonomy Lie algebrah(G; k)and its solvable quotients. In the special case whenG is a commutator-relators group, such pre-sentations were given in [53]. To generalize these results to arbitrary finitely presented groups,we first compute the cup product map∪ : H1(G; k)∧ H1(G; k)→ H2(G; k), using Fox Calculusand Magnus expansion techniques modelled on the approach from [21]. The next result is asummary of Proposition8.9and Theorems9.6, 10.1, and10.5.

Theorem E. Let G be a group with finite presentation〈x1, . . . , xn | r1, . . . , rm〉, and let b=dimH1(G; k).

(1) There exists a groupG with echelon presentation〈x1, . . . , xn | w1, . . . ,wm〉 such thath(G; k) � h(G; k).

(2) The holonomy Lie algebrah(G; k) is the quotient of the freek-Lie algebra with gener-ators y = {y1, . . . , yb} in degree1 by the ideal I generated byκ2(wn−b+1), . . . , κ2(wm),whereκ2 is determined by the Magnus expansion forG.

(3) The solvable quotienth(G; k)/h(G; k)(i) is isomorphic tolie(y)/(I + lie(i)(y)).

Here we say thatG has an ‘echelon presentation’ if the augmented Jacobian matrix of Foxderivatives of this presentation is in row-echelon form. TheoremE yields an algorithm forfinding a presentation for the holonomy Lie algebra of a finitely presented group, and thus, apresentation for the associated graded Lie algebra of a finitely presented, graded-formal group.

1.7. Hilbert series. An important aspect in the study of the graded Lie algebras attached to afinitely generated groupG is the computation of the Hilbert series of these objects. Ifg is sucha graded Lie algebra, andU(g) is its universal enveloping algebra, the Poincare–Birkhoff–Witttheorem expresses the graded ranks ofg in terms of the Hilbert series ofU(g).

In favorable situations, which oftentimes involve the formality notions discussed above, thisapproach permits us to determine the LCS ranksφk(G) = dim grk(G; k) or the Chen ranksθk(G) = dim grk(G/G

′′; k), as well as the holonomy versions of these ranks,φk(G) = dimhk(G; k)andθk(G) = dimhk(G; k)/h′′k (G; k). In this context, the isomorphisms provided by TheoremC,as well as the presentations provided by TheoremE prove to be valuable tools.

Using these techniques, we compute in Section11 the ranksφk(G) andθk(G) for one-relatorgroupsG, while in Section12we compute the whole set of ranks for the fundamental groups ofclosed, orientable Seifert manifolds.

1.8. Further applications. We illustrate our approach with several other classes of finitelypresented groups. We first look at 1-relator groups, whose associated graded Lie algebras werefirst determined by Labute in [30]. We give in Subsections11.1–11.3 presentations for theholonomy Lie algebra and the Chen Lie algebras of a 1-relatorgroup, compute the respectiveHilbert series, and discuss the formality properties of these groups.

It has been known since the pioneering work of W. Massey [46] that fundamental groups oflink complements are not always 1-formal. In fact, as shown by R. Hain in [24], such groupsneed not be graded-formal. However, as shown in [1, 3, 53], if the linking graph is connected,then the link group is graded-formal. Building on work from [17], we give in Subsection11.4an example of a link group which is graded-formal, yet not filtered-formal.


We end in Section12 with a detailed study of fundamental groups of (orientable)Seifertfibered manifolds from a rational homotopy viewpoint. LetM be such a manifold. UsingTheoremE, we find an explicit presentation for the holonomy Lie algebra ofπ1(M). On the otherhand, using the minimal model ofM (as described in [60]), we find a presentation for the MalcevLie algebram(π1(M); k), and use this information to derive a presentation for gr(π1(M); k).As an application, we show that Seifert manifold groups are filtered-formal, and determineprecisely which ones are graded-formal.

This work was motivated in good part by the papers [2, 8] of Etingof et al. on the triangularand quasi-triangular groups, also known as the (upper) purevirtual braid groups. In [66], weapply the techniques developed here to study the formality properties of such groups. Relatedresults for the McCool groups (also known as the welded pure braid groups) and other braid-likegroups will be given in [67, 68].

2. Filtered and graded Lie algebras

In this section we study the interactions between filtered Lie algebras, their completions, andtheir associated graded Lie algebras, mainly as they relateto the notion of filtered formality.

2.1. Graded Lie algebras. We start by reviewing some standard material on Lie algebras,following the exposition from Serre’s book [64] (see also [58, 62, 19]).

Fix a ground fieldk of characteristic 0. Letg be a Lie algebra overk, i.e., ak-vector spacegendowed with a bilinear operation [, ] : g × g→ g satisfying the Lie identities. We say thatg isa graded Lie algebraif g decomposes asg =

⊕i≥1 gi and the Lie bracket sendsgi × g j to gi+ j ,

for all i and j. A morphism of graded Lie algebras is ak-linear mapϕ : g → h which preservesthe Lie brackets and the degrees.

The most basic example of a graded Lie algebra is constructedas follows. LetV a k-vectorspace. The tensor algebraT(V) has a natural Hopf algebra structure, with comultiplication∆and counitε the algebra maps given by∆(v) = v⊗ 1+ 1⊗ v andε(v) = 0, for v ∈ V. ThefreeLie algebraonV is the set of primitive elements, i.e.,

(1) lie(V) = {x ∈ T(V) | ∆(x) = x⊗ 1+ 1⊗ x},

with Lie bracket [x, y] = x⊗ y− y⊗ x and grading induced fromT(V).Now suppose all elements ofV are assigned degree 1 inT(V). Then the inclusionι : lie(V)→

T(V) identifieslie1(V) with T1(V) = V. Furthermore,ιmapslie2(V) to T2(V) = V⊗V by sending[v,w] to v⊗w−w⊗ v for eachv,w ∈ V; we thus may identifylie2(V) � V∧V by sending [v,w]to v∧ w.

A Lie algebrag is said to befinitely generatedif there is an epimorphismϕ : lie(kn) → g forsomen ≥ 1. If, moreover, the Lie idealr = kerϕ is finitely generated as a Lie algebra, theng iscalledfinitely presented.

If g is finitely generated and all the generatorsx1, . . . , xn ∈ lie(kn) can be chosen to havedegree 1, then we sayg is generated in degree1. If, moreover, the Lie idealr is homogeneous,theng is a graded Lie algebra. In particular, ifr is generated in degree 2, then we say the gradedLie algebrag is aquadratic Lie algebra.

2.2. Filtrations. We will be very much interested in this work in Lie algebras endowed with afiltration, usually but not always enjoying an extra ‘multiplicative’ property. At the most basiclevel, afiltration F on a Lie algebrag is a nested sequence of Lie ideals,g = F1g ⊃ F2g ⊃ · · · .

A well-known such filtration is thederived series, Fig = g(i−1), defined byg(0) = g andg(i) =

[g(i−1), g(i−1)] for i ≥ 1. The derived series is preserved by Lie algebra maps. The quotient Lie


algebrasg/g(i) are solvable; moreover, ifg is a graded Lie algebra, all these solvable quotientsinherit a graded Lie algebra structure. The next lemma (which will be used in Section10.2)follows straight from the definitions, using the standard isomorphism theorems.

Lemma 2.1. Let g = lie(V)/r be a finitely generated Lie algebra. Theng/g(i) � lie(V)/(r +lie(V)(i)). Furthermore, ifr is a homogeneous ideal, then this is an isomorphism of gradedLiealgebras.

The existence of a filtrationF on a Lie algebragmakesg into a topological vector space, bydefining a basis of open neighborhoods of an elementx ∈ g to be{x+Fkg}k∈N. The fact that eachbasis neighborhoodFkg is a Lie subalgebra implies that the Lie bracket map [, ] : g × g→ g iscontinuous; thus,g is, in fact, a topological Lie algebra. We say thatg is complete(respectively,separated) if the underlying topological vector space enjoys those properties.

Given an ideala ⊂ g, there is an induced filtration on it, given byFka = Fkg ∩ a. Likewise,the quotient Lie algebra,g/a, has a naturally induced filtration with termsFkg/Fka. Leta be theclosure ofa in the filtration topology. Thena is a closed ideal ofg. Moreover, by the continuityof the Lie bracket, we have that

(2) [a, r] = [a, r].

Finally, if g is complete (or separated), theng/a is also complete (or separated).

2.3. Completions. For eachj ≥ k, there is a canonical projectiong/F jg→ g/Fkg, compatiblewith the projections fromg to its quotient Lie algebrasg/Fkg. Thecompletionof the Lie algebrag with respect to the filtrationF is defined as the limit of this inverse system, i.e.,

(3) g := lim←−−

k

g/Fkg ={(g1, g2, . . . ) ∈

∞∏

i=1

g/Fig∣∣∣ g j ≡ gk modFkg for all j > k

}.

It is readily seen thatg is a Lie algebra, with Lie bracket defined componentwise. Further-more,g has a natural inverse limit filtration,F , given by

(4) Fkg := Fkg = lim←−−i≥k

Fkg/Fig = {(g1, g2, . . . ) ∈ g | gi = 0 for all i < k}.

Note thatFkg = Fkg, and so each term of the filtrationF is a closed Lie ideal ofg. Further-more, the Lie algebrag, endowed with this filtration, is both complete and separated.

Letα : g→ g be the canonical map to the completion. Thenα is a morphism of Lie algebras,preserving the respective filtrations. Clearly, ker(α) =

⋂k≥1 Fkg. Hence,α is injective if and

only if g is separated. Furthermore,α is bijective if and only ifg is complete and separated.

2.4. Filtered Lie algebras. A filtered Lie algebra(over the fieldk) is a Lie algebrag endowedwith ak-vector filtration{Fkg}k≥1 satisfying the ‘multiplicativity’ condition

(5) [Frg,Fsg] ⊆ Fr+sg

for all r, s≥ 1. Obviously, this condition implies that each subspaceFkg is a Lie ideal, and so,in particular,F is a Lie algebra filtration. Let

(6) grF (g) :=⊕

k≥1

Fkg/Fk+1g.

be the associated graded vector space to the filtrationF on g. Condition (5) implies that theLie bracket map ong descends to a map [, ] : grF (g)×grF (g)→ grF (g), which makes grF (g)into a graded Lie algebra, with graded pieces given by decomposition (6).


A morphism of filtered Lie algebras is a linear mapφ : g → h preserving Lie brackets andthe given filtrations,F andG . Such a morphism induces a morphism of associated graded Liealgebras, gr(φ) : grF (g)→ grG (h).

If g is a filtered Lie algebra, then its completion,g, is again a filtered Lie algebra. Indeed, ifF is the given multiplicative filtration ong, andF is the completed filtration ong, thenF alsosatisfies property (5). Moreover, the canonical map to the completion,α : g→ g, is a morphismof filtered Lie algebras. It is readily seen thatα induces isomorphisms

(7) g/Fkg→ g/Fkg,

for eachk ≥ 1, see e.g. [19] From the 5-lemma, we obtain an isomorphism of graded Liealgebras,

(8) gr(α) : grF (g)→ grF (g).

Lemma 2.2. Let φ : g → h be a morphism of complete, separated, filtered Lie algebras,andsupposegr(φ) : grF (g)→ grG (h) is an isomorphism. Thenφ is also an isomorphism.

Proof. The mapφ induces morphismsφk : g/Fkg → h/Gkh for all k ≥ 1. By assumption,the homomorphisms grk(φ) : Fkg/Fk−1g → Gkh/Gk−1h are isomorphisms, for allk > 1. Aneasy induction onk shows that all mapsφk are isomorphisms. Hence, the mapφ : g → h is anisomorphism. By assumption, though,g = g andh = h; henceφ = φ, and we are done. �

Any Lie algebrag comes equipped with a lower central series (lcs) filtration,{Γk(g)}k≥1,defined byΓ1(g) = g andΓk(g) = [Γk−1(g), g] for k ≥ 2. Clearly, this is a multiplicative filtration.Any other such filtration{Fk(g)}k≤1 ong is coarser than this filtration; that is,Γkg ⊆ Fkg, for allk ≥ 1. Any Lie algebra morphismφ : g → h preserves lcs filtrations. Furthermore, the quotientLie algebrasg/Γkg are nilpotent. For simplicity, we shall write gr(g) := grΓ(g) for the associatedgraded Lie algebra andg for the completion ofg with respect to the lcs filtrationΓ. Furthermore,we shall takeΓk = Γk as the canonical filtration ong.

Any graded Lie algebra,g =⊕

i≥1 gi has a canonical decreasing filtration induced by thegrading,Fkg =

⊕i≥k gi . Moreover, ifg is generated in degree 1, then this filtration coincides

with the lcs filtrationΓk(g). In particular, the associated graded Lie algebra with respect toF

coincides withg. In this case, the completion ofg with respect to the lower central series (or,degree) filtration is called thedegree completionof g, and is simply denoted byg. It is readilyseen thatg =

∏i≥1 gi . Therefore, the morphismα : g→ g is injective, and induces isomorphism

ang = grΓ(g). Moreover, ifh is a graded Lie subalgebra ofg, thenh = h and

(9) grΓ(h) = h.

2.5. Filtered formality. We now consider in more detail the relationship between a filtered Liealgebrag and the completion of its associated graded Lie algebra,gr(g).

Lemma 2.3. Letg be a complete, filteredk-Lie algebra. There is then a canonical epimorphismof filteredk-vector spaces,Ψ : g։ gr(g).

Proof. Let {Fk}k≥1 be the filtration ong, and letx be an element ing. Supposex has weightk,that is,x ∈ Fk but x < Fk+1. Define a vector space homomorphismψ : g → gr(g) by ψ(x) = xmodFk+1. Thenψ is surjective and induces a homomorphismψ : g ։ gr(g) which preservesthe respective filtrations. Taking completions, we obtain the desired morphism,Ψ := ψ : g ։gr(g). �


As we shall see in Example7.7, the mapΨ from above need not be a Lie algebra morphism.This brings us to a definition which will play a key role in the sequel.

Definition 2.4. A complete, filtered Lie algebrag is calledfiltered-formalif there is a filteredLie algebra isomorphismg � gr(g) which induces the identity on associated graded algebras.

This notion appears in the work of Bezrukavnikov [5] and Hain [25], as well as in the workof Calaque–Enriquez–Etingof [8] under the name of ‘formality’, and in the work of Lee [38],under the name of ‘weak-formality’. The reasons for our choice of terminology will becomemore apparent in Section5.

It is easy to construct examples of Lie algebras that enjoy this property. For instance, ifm = g is the completion of a finitely-generated Lie algebrag which admits a homogeneouspresentation, thenm is filtered-formal. In particular, ifg =

⊕i≥1 gi is a finitely generated

graded Lie algebra generated in degree 1, theng =∏

i≥1 gi a filtered-formal Lie algebra.

Lemma 2.5. Let g be a complete, filtered Lie algebra, and leth be a graded Lie algebra. Ifthere is a Lie algebra isomorphismg � h preserving filtrations, theng is filtered-formal.

Proof. By assumption, there exists a filtered Lie algebra isomorphismΦ : g → h. The mapΦinduces a graded Lie algebra isomorphism, gr(Φ) : gr(g) → h. In turn, the mapΨ := (gr(Φ))−1

induces an isomorphismΨ : h → gr(g) of completed Lie algebras. Hence, the compositionΨ ◦ Φ : g→ gr(g) is an isomorphism of filtered Lie algebras inducing the identity on gr(g). �

Corollary 2.6. Supposem is a filtered-formal Lie algebra. There exists then a graded Liealgebrag such thatm is isomorphic tog =

∏i≥1 gi .

2.6. Products and coproducts.The category of Lie algebras admits both products and co-products. We conclude this section by showing that filtered formality behaves well with respectto these operations.

Lemma 2.7. Letm andn be two filtered-formal Lie algebras. Thenm×n is also filtered-formal.

Proof. By assumption, there exist graded Lie algebrasg andh such thatm � g =∏

i≥1 gi andn � h =

∏i≥1 hi . We then have

(10) m × n �(∏

i≥1

gi

)×

(∏

i≥1

hi

)=

∏

i≥1

(gi × hi) = g × h.

Hence,m × n is filtered-formal. �

Now let ∗ denote the usual coproduct (or, free product) of Lie algebras, and let ˆ∗ be thecoproduct in the category of complete, filtered Lie algebras. By definition,

(11) m ∗ n = m ∗ n = lim←−−

k

(m ∗ n)/Γk(m ∗ n).

We refer to Lazarev and Markl [37] for a detailed study of this notion.

Lemma 2.8. Letm andn be two filtered-formal Lie algebras. Thenm ∗ n is also filtered-formal.

Proof. As before, writem = g andn = h, for some graded Lie algebrasg andh. The canon-ical inclusions,α : g → m andβ : h → n, induce a monomorphism of filtered Lie algebras,α ∗ β : g ∗ h → m ∗ n. Using [37, (9.3)], we infer that the induced morphism between asso-ciated graded Lie algebras, gr(α ∗ β) : gr(g ∗ h) → gr(m ∗ n), is an isomorphism. Lemma2.2now implies thatα ∗ β is an isomorphism of filtered Lie algebras, thereby verifying the filtered-formality ofm ∗ n. �


3. Graded algebras and Koszul duality

The notions of graded and filtered algebras are defined completely analogously for an (as-sociative) algebraA: the multiplication map is required to preserve the grading, respectivelythe filtration onA. In this section we discuss several relationships between Lie algebras andassociative algebras, focussing on the notion of quadraticand Koszul algebras.

3.1. Universal enveloping algebras.Given a Lie algebrag over a fieldk of characteristic 0,let U(g) be its universal enveloping algebra. This is the filtered algebra obtained as the quotientof the tensor algebraT(g) by the (two-sided) idealI generated by all elements of the forma⊗ b− b⊗ a− [a, b] with a, b ∈ g. By the Poincare–Birkhoff–Witt theorem, the canonical mapι : g → U(g) is an injection, and the induced map, Sym(g) → gr(U(g)), is an isomorphism ofgraded (commutative) algebras.

Now supposeg is a finitely generated, graded Lie algebra. ThenU(g) is isomorphic (asa graded vector space) to a polynomial algebra in variables indexed by bases for the gradedpieces ofg, with degrees set accordingly. Hence, its Hilbert series isgiven by

(12) Hilb(U(g), t) =∏

i≥1

(1− ti)−φi (g),

whereφi(g) = dimk(gi). For instance, ifg = lie(V) is the free Lie algebra on a finite-dimensionalvector spaceV with all generators in degree 1, thenφi(g) = 1

i

∑d|i µ(d) · ni/d, wheren = dimV

andµ : N→ {−1, 0, 1} is the Mobius function.Finally, supposeg = lie(V)/r is a finitely presented, graded Lie algebra, with generatorsin

degree 1 and relation idealr generated by homogeneous elementsg1, . . . , gm. ThenU(g) is thequotient ofT(V) by the two-sided ideal generated byι(g1), . . . , ι(gm), whereι : lie(V) → T(V)is the canonical inclusion. In particular, ifg is a quadratic Lie algebra, thenU(g) is a quadraticalgebra.

3.2. Quadratic algebras. Now let A be a gradedk-algebra. We will assume throughout thatA is non-negatively graded, i.e.,A =

⊕i≥0 Ai , and connected, i.e.,A0 = k. Every such algebra

may be realized as the quotient of a tensor algebraT(V) by a homogeneous, two-sided idealI .We will further assume that dimV < ∞.

An algebraA as above is said to bequadraticif A1 = V and the idealI is generated in degree2, i.e., I = 〈I2〉, whereI2 = I ∩ (V ⊗ V). Given a quadratic algebraA = T(V)/I , identifyV∗ ⊗ V∗ � (V ⊗ V)∗, and define thequadratic dualof A to be the algebra

(13) A! = T(V∗)/I⊥,

whereI⊥ ⊂ T(V∗) is the ideal generated by the vector subspaceI⊥2 := {α ∈ V∗ ⊗V∗ | α(I2) = 0}.Clearly,A! is again a quadratic algebra, and (A!)! = A.

For any graded algebraA = T(V)/I , we can define a quadrature closureA = T(V)/〈I2〉.

Proposition 3.1. Let g be a finitely generated graded Lie algebra generated in degree1. Thereis then a unique, functorially defined quadratic Lie algebra, g, such that U(g) = U(g).

Proof. Supposeg has presentationlie(V)/r. ThenU(g) has a presentationT(V)/(ι(r)). Setg = lie(V)/〈r2〉, wherer2 = r ∩ lie2(V); thenU(g) has presentationT(V)/〈ι(r2)〉. One can seethatι(r2) = ι(r) ∩ V ⊗ V. �

A commutative graded algebra(for short, a cga) is a gradedk-algebra as above, which inaddition is graded-commutative, i.e., ifa ∈ Ai andb ∈ A j , thenab= (−1)i jba. If all generators


of A are in degree 1, thenA can be written asA =∧

(V)/J, where∧

(V) is the exterior algebra onthek-vector spaceV = A1, andJ is a homogeneous ideal in

∧(V) with J1 = 0. If, furthermore,

J is generated in degree 2, thenA is a quadratic cga. The next lemma follows straight from thedefinitions.

Lemma 3.2. Let W⊂ V∧V be a linear subspace, and let A=∧

(V)/〈W〉 be the correspondingquadratic cga. Then A! = T(V∗)/〈ι(W∨)〉, where

(14) W∨ := {α ∈ V∗ ∧ V∗ | α(W) = 0} =W⊥ ∩ (V∗ ∧ V∗),

andι : V∗ ∧ V∗ → V∗ ⊗ V∗ is the inclusion map, given by x∧ y 7→ x⊗ y− y⊗ x.

For instance, ifA =∧

(V), thenA! = Sym(V∗). Likewise, if A =∧

(V)/〈V ∧ V〉 = k ⊕ V,thenA! = T(V∗).

3.3. Holonomy Lie algebras. Let A be a commutative graded algebra. Recall we are assumingthat A0 = k and dimA1 < ∞. Because of graded-commutativity, the multiplication mapA1 ⊗

A1 → A2 factors through a linear mapµA : A1 ∧ A1 → A2. Dualizing this map, and identifying(A1 ∧ A1)∗ � A∗1 ∧ A∗1, we obtain a linear map,

(15) ∂A = (µA)∗ : A∗2→ A∗1 ∧ A∗1.

Finally, identifyA∗1 ∧ A∗1 with lie2(A∗1) via the mapx∧ y 7→ [x, y].

Definition 3.3. Theholonomy Lie algebraof A is the quotient

(16) h(A) = lie(A∗1)/〈im ∂A〉

of the free Lie algebra onA∗1 by the ideal generated by the image of∂A under the above identi-fication. Alternatively,

(17) h(A) = lie(A∗1)/〈ker(µA)∨〉.

By construction,h(A) is a quadratic Lie algebra. Moreover, this construction isfunctorial: ifϕ : A→ B is a morphism of cga’s as above, the induced map,lie(ϕ∗1) : lie(B∗1)→ lie(A

∗1), factors

through a morphism of graded Lie algebras,h(ϕ) : h(B) → h(A). Moreover, ifϕ is injective,thenh(ϕ) is surjective.

Clearly, the holonomy Lie algebrah(A) depends only on the information encoded in themultiplication mapµA : A1 ∧ A1 → A2. More precisely, letA be thequadratic closureof Adefined as

(18) A =∧

(A1)/〈K〉,

whereK = ker(µA) ⊂ A1 ∧ A1. Then A is a commutative, quadratic algebra, which comesequipped with a canonical homomorphismq: A → A, which is an isomorphism in degree 1and a monomorphism in degree 2. It is readily verified that theinduced morphism betweenholonomy Lie algebras,h(A)→ h(A), is an isomorphism.

The following proposition is a slight generalization of [56, Lemma 4.1].

Proposition 3.4. Let A be a commutative graded algebra. Then U(h(A)) is a quadratic algebra,and U(h(A)) = A!.

Proof. By the above,A =∧

(A1)/〈K〉, whereK = 〈ker(µA)〉. On the other hand, by (17) wehave thath(A) = lie(A∗1)/〈K

∨〉. Hence, by Lemma3.2, U(h(A)) = T(V∗)/〈ι(K∨)〉 = A!. �

Combining Propositions3.1and3.4, we can see the relations between the quadratic closureof a Lie algebra and the holonomy Lie algebra.


Corollary 3.5. Let g be a finitely generated graded Lie algebra generated in degree 1. Then

h(U(g)

!)= g.

Work of Lofwall [41] yields another interpretation of the universal enveloping algebra of theholonomy Lie algebra.

Proposition 3.6 ([41]). Let Ext1A(k, k) =⊕

i≥0 ExtiA(k, k)i be the linear strand in the Yonedaalgebra of A. Then U(h(A)) � Ext1A(k, k).

In particular, the graded ranks of the holonomy Lie algebrah = h(A) are given by∏

n≥1(1−tn)φn(h) =

∑i≥0 bii ti , wherebii = dimk ExtiA(k, k)i.

The next proposition shows that every quadratic Lie algebracan be realized as the holonomyLie algebra of a (quadratic) algebra.

Proposition 3.7. Let g be a quadratic Lie algebra. There is then a commutative quadraticalgebra A such thatg = h(A).

Proof. By assumption,g has a presentation of the formlie(V)/〈W〉, whereW is a linear subspaceof V ∧ V. DefineA =

∧(V∗)/〈W∨〉. Then, by (17),

(19) h(A) = lie((V∗)∗)/〈(W∨)∨〉 = lie(V)/〈W〉,

and this completes the proof. �

3.4. Koszul algebras. Any connected, graded algebraA =⊕

i≥0 Ai has a free, gradedA-resolution of the trivialA-modulek,

(20) · · ·ϕ3 // Ab2

ϕ2 // Ab1ϕ1 // A // k .

Such a resolution is said to beminimalif all the nonzero entries of the matricesϕi have positivedegrees.

A Koszul algebrais a graded algebra for which the minimal graded resolution of k is linear,or, equivalently, ExtA(k, k) = Ext1A(k, k). Such an algebra is always quadratic, but the converseis far from true. IfA is a Koszul algebra, then the quadratic dualA! is also a Koszul algebra, andthe following ‘Koszul duality’ formula holds:

(21) Hilb(A, t) · Hilb(A! ,−t) = 1.

Furthermore, ifA is a graded algebra of the formA = T(V)/I , whereI is an ideal admittinga (noncommutative) quadratic Grobner basis, thenA is a Koszul algebra (see [23]).

Corollary 3.8. Let A be a connected, commutative graded algebra. IfA is a Koszul algebra,thenHilb(A,−t) · Hilb(U(h(A)), t) = 1.

Example 3.9. Consider the quadratic algebraA =∧

(u1, u2, u3, u4)/(u1u2 − u3u4). Clearly,Hilb(A, t) = 1+ 4t + 5t2. If A were Koszul, then formula (21) would give Hilb(A! , t) = 1+ 4t +11t2 + 24t3 + 41t4 + 44t5 − 29t6 + · · · , which is impossible.

Example 3.10. The quasitriangular Lie algebraqtrn defined in [2] is generated byxi j , 1 ≤ i ,j ≤ n with relations [xi j , xik] + [xi j , x jk] + [xik, x jk] = 0 for distincti, j, k and [xi j , xkl] = 0 fordistinct i, j, k, l. The Lie algebratrn is the quotient Lie algebra ofqtrn by the ideal generated byxi j + x ji for distincti , j. In [2], Bartholdi, Enriquez, Etingof, and Rains show that the quadraticdual algebrasU(qtrn)! andU(trn)! are Koszul, and compute their Hilbert series. They also statethat neitherqtrn nor trn is filtered-formal forn ≥ 4, and sketch a proof of this fact. We willprovide a detailed proof in [66].


4. Groups, Lie algebras, and graded formality

We now turn to finitely generated groups, and to two graded Liealgebras attached to suchgroups, with special emphasis on the relationship between these Lie algebras, leading to thenotion of graded formality.

4.1. Central filtrations on groups. We start with some general background on lower centralseries and the associated graded Lie algebra of a group. For more details on this classical topic,we refer to Lazard [36] and Magnus–Karrass–Solitar [43].

Let G be a group. For elementsx, y ∈ G, let [x, y] = xyx−1y−1 be their group commutator.Likewise, for subgroupsH,K < G, let [H,K] be the subgroup ofG generated by all commuta-tors [x, y] with x ∈ H, y ∈ K.

A (central) filtration on the groupG is a decreasing sequence of subgroups,G = F1G >

F2G > F3G > · · · , such that [FrG,FsG] ⊂ Fr+sG. It is readily verified that, for eachk > 1,the groupFk+1G is a normal subgroup ofFkG, and the quotient group grF

k (G) = FkG/Fk+1Gis abelian. As before, letk be a field of characteristic 0. The direct sum

(22) grF (G; k) =⊕

k≥1

grFk (G) ⊗Z k

is a graded Lie algebra overk, with Lie bracket induced from the group commutator: Ifx ∈ FrGandy ∈ FsG, then [x+Fr+1G, y+Fs+1G] = xyx−1y−1 +Fr+s+1G. We can view grF (−; k) asa functor from groups to gradedk-Lie algebras.

Let H be a normal subgroup ofG, and letQ = G/H be the quotient group. Define filtrationson H andQ by FkH = FkG ∩ H andF kQ = FkG/FkH, respectively. We then have thefollowing classical result of Lazard.

Proposition 4.1(Theorem 2.4 in [36]). The canonical projection G։ G/H induces a naturalisomorphism of graded Lie algebras,

grF (G)/ grF (H) ≃ // grF (G/H) .

4.2. The associated graded Lie algebra.Any groupG comes endowed with the lower centralseries (lcs) filtration{ΓkG}k≥1, defined inductively byΓ1G = G and

(23) Γk+1G = [ΓkG,G].

If ΓkG , 1 butΓk+1G = 1, thenG is said to be ak-step nilpotent group. In general, though, thelcs filtration does not terminate.

The Lie algebra gr(G; k) = grΓ(G; k) is called theassociated graded Lie algebra(overk) ofthe groupG. For instance, ifF = Fn is a free group of rankn, then gr(F; k) is the free gradedLie algebralie(kn). A group homomorphismf : G1 → G2 induces a morphism of graded Liealgebras, gr(f ; k) : gr(G1; k) → gr(G2; k); moreover, if f is surjective, then gr(f ; k) is alsosurjective.

For eachk ≥ 2, the factor groupG/Γk(G) is the maximal (k− 1)-step nilpotent quotient ofG.The canonical projectionG → G/Γk(G) induces an epimorphism gr(G, k) → gr(G/Γk(G), k),which is an isomorphism in degreess< k.

From now on, unless otherwise specified, we will assume that the groupG is finitely gen-erated. That is, there is a free groupF of finite rank, and an epimorphismϕ : F ։ G. LetR = ker(ϕ); thenG = F/R is called a presentation forG. Note that the induced morphismgr(ϕ; k) : gr(F; k) → gr(G; k) is surjective. Thus, gr(G; k) is a finitely-generated Lie algebra,with generators in degree 1. We will omit the coefficient fieldk if there is no confusion.


Let H ⊳G be a normal subgroup, and letQ = G/H. If Γr H = ΓrG∩H is the induced filtrationon H, it is readily seen that the filtrationΓr Q = ΓrG/ΓrH coincides with the lcs filtration onQ.Hence, by Proposition4.1,

(24) gr(Q) � gr(G)/ grΓ(H).

Now supposeG = H ⋊ Q is a semi-direct product of groups. In general, there is not muchof a relation between the respective associated graded Lie algebras. Nevertheless, we have thefollowing well-known result of Falk and Randell [20], which shows that gr(G) = gr(H) ⋊ gr(Q)for ‘almost-direct’ products of groups.

Theorem 4.2([20]). Let G = H ⋊ Q be a semi-direct product of groups, and suppose Q actstrivially on Hab. Then the filtrations{Γr H}r≥1 and {Γr H}r≥1 coincide, and there is a split exactsequence of graded Lie algebras,

0 // gr(H) // gr(G) // gr(Q) // 0 .

4.3. The holonomy Lie algebra. The holonomy Lie algebra of a finitely generated group wasintroduced by K.-T. Chen [13] and Kohno [29], and further studied in [45, 53].

Definition 4.3. Let G be a finitely generated group. Theholonomy Lie algebraof G is theholonomy Lie algebra of the cohomology ringA = H∗(G; k), that is,

(25) h(G; k) = lie(H1(G, k))/〈im∂G〉,

where∂G is the dual to the cup-product map∪G : H1(G, k) ∧ H1(G, k)→ H2(G, k).

By construction,h(G; k) is a quadratic Lie algebra. Iff : G1 → G2 is a group homomor-phism, then the induced homomorphism in cohomology,f ∗ : H1(G2, k) → H1(G1, k) yields amorphism of graded Lie algebras,h( f ; k) : h(G1; k) → h(G2, k). Moreover, if f is surjective,thenh( f ; k) is also surjective.

In the definition of the holonomy Lie algebra ofG, we used the cohomology ring of a clas-sifying spaceK(G, 1). As the next lemma shows, we may replace this space by any otherconnected CW-complex with the same fundamental group.

Lemma 4.4. Let X be a connected CW-complex withπ1(X) = G. Thenh(H∗(X, k)) � h(G; k).

Proof. We may construct a classifying space forG by adding cells of dimension 3 and higher toX in a suitable way. The inclusion map,j : X→ K(G, 1), induces a map on cohomology rings,j∗ : H∗(K(G, 1);k)→ H∗(X; k), which is an isomorphism in degree 1 and an injection in degree2. Consequently,j2 restricts to an isomorphism from im(∪G) to im(∪X). Taking duals, we findthat im(∂X) = im(∂G). The conclusion follows. �

In particular, if KG is the 2-complex associated to a presentation ofG, then h(G; k) �h(H∗(KG, k)). Let φn(G) := dimhn(G; k) be the dimensions of the graded pieces of the holo-nomy Lie algebra ofG. The next corollary is an algebraic version of the lcs formula from [56],but with no formality assumption.

Corollary 4.5. Let X be a connected CW-complex withπ1(X) = G, let A = H∗(X; k) be itscohomology algebra, and letA be the quadratic closure of A. Then

∏n≥1(1−tn)φn(G) =

∑i≥0 bii ti ,

where bii = dim ExtiA(k, k)i. Moreover, ifA is a Koszul algebra, then∏

n≥1

(1− tn)φn = Hilb(A,−t).


Proof. The first claim follows from Lemma4.4, the Poincare–Birkhoff–Witt formula (12), andLofwall’s formula from Proposition3.6. The second claim follows from the Koszul dualityformula stated in Corollary3.8. �

4.4. A comparison map. Once again, letG be a finitely generated group. Although the nextlemma is known, we provide a proof, both for the sake of completeness, and for later use.

Lemma 4.6([45, 53]). There exists a natural epimorphism of gradedk-Lie algebras,

(26) ΦG : h(G; k) // // gr(G; k) ,

inducing isomorphisms in degrees1 and2.

Proof. As first noted by Sullivan [69] in a particular case, and proved by Lambe [34] in general,there is a natural exact sequence

(27) 0 // (Γ2G/Γ3G⊗ k)∗β // H1(G; k) ∧ H1(G; k) ∪ // H2(G; k) ,

whereβ is the dual of Lie bracket product. In particular, im(∂G) = ker(β∗).Recall that the associated graded Lie algebra gr(G; k) is generated by its degree 1 piece,

H1(G; k) � gr1(G) ⊗ k. Hence, there is a natural epimorphism of gradedk-Lie algebras,

(28) ϕG : lie(H1(G; k)) // // gr(G; k) ,

restricting to the identity in degree 1, and the Lie bracket map [, ] :∧2 gr1(G; k) → gr2(G; k)

in degree 2. In the exact sequence (27), the image of∂G coincides with the kernel of the Liebracket map. This completes the proof. �

Corollary 4.7. Let V = H1(G; k). Suppose the associated graded Lie algebrag = gr(G; k)has presentationlie(V)/r. Then the holonomy Lie algebrah(G; k) has presentationlie(V)/〈r2〉,wherer2 = r ∩ lie2(V). Furthermore, if A= U(g), thenh(G; k) = h

(A!).

Proof. Taking the dual of the exact sequence (28), we find that im(∂G) = ker(β∗), whereβ : V ∧V → lie2(V) is the Lie bracket inlie(V). Hence,〈r2〉 = 〈im(∂G)〉 as ideals oflie(V); thus,h(G; k) = lie(V)/〈r2〉. The last claim follows from Corollary3.5. �

Recall we denote byφn(G) andφn(G) the dimensions on then-th graded pieces of gr(G; k)andh(G; k), respectively. By Lemma4.6, φn(G) ≥ φn(G), for all n ≥ 1, and equality alwaysholds forn ≤ 2. Nevertheless, these inequalities can be strict forn ≥ 3.

As a quick application, let us compare the holonomy Lie algebras ofG and its nilpotentquotients.

Proposition 4.8. Let G be a finitely generated group. Then,

(29) h(G/ΓkG; k) =

h(G; k)/h(G; k)′ for k = 2,

h(G; k) for k ≥ 3.

Proof. The casek = 2 is trivial, so let us assumek ≥ 3. By a previous remark, the pro-jection G → G/Γk(G) induces an isomorphism gr2(G, k) → gr2(G/Γk(G), k). Furthermore,H1(G; k) � H1(G/Γk(G); k). Using now the dual of the exact sequence (27), we see thatim(∂G) = im(∂G/Γk(G)). The desired conclusion follows. �


4.5. Graded-formality. We continue our discussion of the associated graded and holonomyLie algebras of a finitely generated group with a formality notion that will be important in thesequel.

Definition 4.9. A finitely-generated groupG is graded-formal(overk) if the canonical projec-tionΦG : h(G; k)։ gr(G; k) is an isomorphism of graded Lie algebras.

This notion was introduced by Lee in [38], where it is called graded 1-formality. It is readilyseen that the definition is independent of the choice of coefficient fieldk of characteristic 0.

Lemma 4.10.A finitely-generated group G is graded-formal if and only ifgr(G; k) is quadratic.

Proof. The forward implication is immediate. So assume gr(G; k) is quadratic, that is, it admitsa presentation of the formlie(V)/〈U〉, whereV is ak-vector space in degree 1 andU is ak-vectorsubspace oflie2(V). In particular,V = gr1(G; k) = H1(G; k).

From the exact sequence (27), we see that the image of∂G coincides with the kernel of theLie bracket map [, ] :

∧2 gr1(G; k) → gr2(G; k), which can be identified withU. Hence thesurjectionϕG : lie(Gk)։ gr(G; k) induces an isomorphismΦG : h(G; k) ≃−→ gr(G; k). �

In fact, the groupG is graded-formal if and only ifφn(G) = φn(G) for all n ≥ 3.

4.6. Split injections. We are now in a position to state and prove the main result of this section,which proves the first part of TheoremA from the Introduction.

Theorem 4.11. Let G be a finitely generated group. Suppose there is a split monomorphismι : K → G. If G is a graded-formal group, then K is also graded-formal.

Proof. In view of our hypothesis, we have an epimorphismσ : G ։ K such thatσ ◦ ι = id.In particular,K is also finitely generated. Furthermore, the induced mapsh(ι) and gr(ι) are alsoinjective.

Let π : F ։ G be a presentation forG. There is then an induced presentation forK, given bythe compositionσπ : F ։ K. By Lemma4.6, there exist epimorphismsΦ1 andΦ making thefollowing diagram commute.

(30) h(G1; k)� _

h(ι)

��

Φ1 // // gr(K; k)� _

gr(ι)

��h(G; k)

Φ // // gr(G; k)

If the groupG is graded-formal, thenΦ is an isomorphism of graded Lie algebras. Hence,the epimorphismΦ1 is also injective, and soK is a graded-formal. �

Theorem 4.12. Let G = K ⋊ Q be a semi-direct product of finitely generated groups, andsuppose G is graded-formal. Then:

(1) The group Q is graded-formal.(2) If, moreover, Q acts trivially on Kab, then K is also graded-formal.

Proof. The first assertion follows at once from Theorem4.11. So assumeQ acts trivially onKab.By Theorem4.2, there exists a split exact sequence of graded Lie algebras,which we record in


the top row of the next diagram.

(31) 0 // gr(K; k) // gr(G; k)rr

// gr(Q; k)rr

// 0

h(K; k) //

OOOO

h(G; k)ss

//

�

OO

h(Q; k) //ss

�

OO

0 .

Let ι : K → G be the inclusion map. By the above, we have an epimorphism,σ : gr(G; k)→gr(K; k) such thatσ ◦ gr(ι) = id. Consequently, gr(K; k) is finitely generated.

By Corollary4.7, the mapσ induces a morphism ¯σ : h(G; k)→ h(G; k) such that ¯σ◦h(ι) = id.Consequently,h(ι) is injective. Therefore, the morphismh(K; k) → gr(K; k) is also injective.Hence,K is graded-formal. �

If the hypothesis of Theorem4.12, part (2) does not hold, the subgroupK may not be graded-formal, even when the groupG is 1-formal. We illustrate this phenomenon with an exampleadapted from [54].

Example 4.13.Let K = 〈x, y | [x, [x, y]] , [y, [x, y]] 〉 be the discrete Heisenberg group. Considerthe semidirect productG = K ⋊φ Z, defined by the automorphismφ : K → K given byx→ y,y→ xy. We have thatb1(G) = 1, and soG is 1-formal, yetK is not graded-formal.

4.7. Products and coproducts.We conclude this section with a discussion of the functors grandh and the notion of graded formality behave with respect to products and coproducts.

Lemma 4.14([39, 55]). The functorsgr and h preserve products and coproducts, that is, wehave the following natural isomorphisms of graded Lie algebras,

gr(G1 ×G2; k) � gr(G1; k) × gr(G2; k)

gr(G1 ∗G2; k) � gr(G1; k) ∗ gr(G2; k),and

h(G1 ×G2; k) � h(G1; k) × h(G2; k)

h(G1 ∗G2; k) � h(G1; k) ∗ h(G2; k).

Proof. The first statement on the gr(−) functor is well-known, while the second statement is themain theorem from [39]. The statements regarding theh(−) functor can be found in [55]. �

Regarding graded-formality, we have the following result,which sharpens and generalizesLemma 4.5 from [57], and proves the first part of TheoremB from the Introduction.

Proposition 4.15. Let G1 and G2 be two finitely generated groups. Then, the following condi-tions are equivalent.

(1) G1 and G2 are graded-formal.(2) G1 ∗G2 is graded-formal.(3) G1 ×G2 is graded-formal.

Proof. Since there exist split injections fromG1 andG2 to the productG1 ×G2 and coproductG1∗G2, Theorem4.11shows that implications (2)⇒(1) and (3)⇒(1) hold. Implications (1)⇒(2)and (1)⇒(3) follow from Lemma4.14and the naturality of the mapΦ from (26). �

5. Malcev Lie algebras and filtered formality

In this section we consider the Malcev Lie algebra of a finitely generated group, and studythe ensuing notions of filtered formality and 1-formality.


5.1. Prounipotent completions and Malcev Lie algebras.Once again, letG be a finitely-generated group, and let{ΓkG}k≥1 be its lcs filtration. The successive quotients ofG by thesenormal subgroups form a tower of nilpotent groups,

(32) · · · // G/Γ4G // G/Γ3G // G/Γ2G = Gab .

Let k be a field of characteristic 0. It is possible to replace each nilpotent quotientNk =

G/ΓkG by Nk ⊗ k, the (rationally defined) nilpotent Lie group associated toNk/tors(Nk) via aprocedure which will be discussed further in§7.1. The corresponding inverse limit,

(33) M(G; k) = lim←−−

k

((G/ΓkG) ⊗ k),

is a prounipotent Lie group overk, called theprounipotent completion, or Malcev completionof G. The (pronilpotent) Lie algebra ofM(G; k) is called theMalcev Lie algebraof G, and isdenoted bym(G; k). By construction,m(G; k) is a complete, separated, filtered Lie algebra.

In [62], Quillen gave a different construction of this Lie algebra. The group-algebrakG has anatural Hopf algebra structure, with comultiplication∆ : kG⊗ kG→ kG given by∆(g) = g⊗ gfor g ∈ G, and counit the augmentation mapε : kG → k. The powers of the augmentationideal I = kerε form a descending filtration ofkG by two-sided ideals; letkG = lim

←−−rkG/I r

be the completion of the group-algebra with respect to this filtration. An elementx ∈ kG iscalled primitive if ∆x = x⊗1 + 1⊗x. The set of all primitive elements inkG, with bracket[x, y] = xy− yx, and endowed with the induced filtration, is a Lie algebra, isomorphic to theMalcev Lie algebra ofG,

(34) m(G; k) � Prim(kG

).

The filtration topology onkG is a metric topology; hence, the filtration topology onm(G; k)is also metrizable, and thus separated. For Malcev Lie algebram(G; k), we only deal with theinduced inverse limit filtration. Hence, we can omit the notion for the filtration and denote theassociated graded Lie algebra ofm(G; k) by gr(m(G; k)). Finally, as shown by Quillen in [61],gr(m(G; k)) is isomorphic to the associated graded Lie algebra ofG, i.e.,

(35) gr(m(G; k)) � gr(G, k).

In particular, gr(m(G; k)) is generated in degree 1.Alternatively, the Malcev Lie algebra ofG can be obtained from the first stage of the minimal

modelM =M (APL(K)⊗k), whereAPL(K) is the Sullivan model of rational polynomial forms,andK is a simplicial complex approximation ofK(G, 1), see [70, 10]. Likewise, if (A, d) isa quadratic cdga quasi-isomorphic toM , thenm(G; k) is isomorphic to the completion withrespect to the LCS filtration of the Lie algebra dual to (A, d), see [5, 58, 6, 4]. Finally, if G is afinitely presented group, presentations form(G; k) can be found in [48, 52].

5.2. One-formality. We now come to a crucial notion in non-simply-connected rational ho-motopy theory. LetX be a space having the homotopy type of a connected simplicialcomplexK with finite 1-skeleton. Such a space is said to beformal (in the sense of Sullivan [70]) ifthe modelAPL(K) can be connected via a zig-zag of quasi-isomorphisms to thecohomologyalgebraH∗(X,Q), endowed with the zero differential. If each of the morphisms connecting thetwo cdga’s induces an isomorphism in degree 1 and a monomorphism in degree 2, we say thatXis merely 1-formal. For 2-dimensional CW-complexes, the notions of formality and 1-formalitycoincide (see [42]), but in higher dimensions, full formality is a much stronger condition.


As noted in [54], the 1-formality property of a spaceX depends only on its fundamentalgroup,G = π1(X). Indeed, if f : X → K(G, 1) is a classifying map, then the induced homo-morphism in cohomology is an isomorphism in degree 1 and a monomorphism in degree 2. Asshown in [70], there is an equivalent, purely group-theoretic definition of 1-formality of a group.

Definition 5.1. A finitely-generated groupG is called 1-formal(overk) if its Malcev Lie algebram(G; k) is isomorphic to the completion of the holonomy Lie algebrah(G; k).

Lemma 5.2([54]). Let G be a finitely-generated group. The group G is1-formal if and only ifthe Malcev Lie algebra of G is isomorphic to the degree completion of a rational quadratic Liealgebra, as filtered Lie algebras.

For instance, letF be a finitely generated free group. Then the Malcev Lie algebram(F; k) isisomorphic tolie(Fk), the completed free Lie algebra onFk = H1(F; k). Hence,F is 1-formal.

Other well-known examples of 1-formal groups include the right-angled Artin groups, thepure braid groupsPn, and the pure braid groups of surfaces of genus greater than 1[5, 25].Further examples of 1-formal groups are the fundamental groups of compact Kahler manifolds[16], and the fundamental groups of complements of complex algebraic hypersurfaces [29].

Example 5.3. In [2], Bartholdi, Enriquez, Etingof, and Rains consider two infinite families ofgroups. The first are the quasitriangular groups QTrn, which have presentations with generatorsxi j (1 ≤ i , j ≤ n), and relationsxi j xikx jk = x jkxikxi j andxi j xkl = xklxi j for distinct i, j, k, l. Thesecond are the triangular groups Trn, each of which is the quotient of QTrn by the relations ofthe formxi j = x ji for i , j. As shown in [38], the groups QTrn and Trn are all graded-formal.On the other hand, as indicated in [2], these groups are non-1-formal, for alln ≥ 4. A detailedproof of this fact will be given in [66].

5.3. Massey products.A well-known obstruction to 1-formality is provided by the higher-order Massey products (introduced in [46]). For our purposes, we will discuss here only tripleMassey products of degree 1 cohomology classes.

Let γ1, γ2 andγ3 be cocycles of degrees 1 in the (singular) chain complexC∗(G; k), withcohomology classesui = [γi ] satisfyingu1 ∪ u2 = 0 andu2 ∪ u3 = 0. That is, we assume thereare 1-cochainsγ12 andγ23 such thatdγ12 = γ1 ∪ γ2 anddγ23 = γ2 ∪ γ3. It is readily seen thatthe 2-cochainω = γ12 ∪ γ3 + γ1 ∪ γ23 is, in fact, a cocycle. The set of all cohomology classes[ω] obtained in this way is theMassey triple product〈u1, u2, u3〉 of the classesu1, u2 andu3.Due to the ambiguity in the choice ofγ12 andγ23, the Massey triple product〈u1, u2, u3〉 is arepresentative of the coset

(36) H2(G; k)/(u1∪ H1(G; k) + H1(G; k) ∪ u3).

In [59], Porter gave a topological method for computing cup products products and Masseyproducts inH2(G; k). Building on work of Dwyer [18], Fenn and Sjerve gave in [21] anothermethod for computing these products in the second cohomology of a commutator-relators group,directly from a presentation of the group. We will briefly review the latter method in Remark9.8, and use it in the computations from Examples5.7, 11.10, and11.13.

If a groupG is 1-formal, then all triple Massey products vanish in the quotientk-vector space(36). However, ifG is only graded-formal, these Massey products need not vanish. As we shallsee in Example11.10, even a one-relator groupG may be graded-formal, yet not 1-formal.

5.4. Filtered formality. Next, we define the notion of filtered formality (also known asweakformality [38]) for groups, based on the notion of filtered formality for Lie algebras, which wasintroduced in Definition2.4.


Definition 5.4. A finitely-generated groupG is said to befiltered-formal(overk) if its MalcevLie algebram(G, k) is filtered-formal.

Clearly, the definition is independent of the choice of coefficient fieldk of characteristic 0.Here is a more direct way to think of this notion.

Lemma 5.5. A finitely-generated group G is filtered-formal if and only ifm(G, k) � gr(G, k).

Proof. We know from Quillen’s isomorphism (35) that gr(m(G; k)) � gr(G, k). The forwardimplication follows straight from the definitions, while the backward implication follows fromLemma2.5. �

The next result pulls together the various formality notions for groups, and establishes thebasic relationship among them.

Proposition 5.6. A finitely-generated group G is1-formal if and only if G is graded-formal andfiltered-formal.

Proof. First supposeG is 1-formal. Thenm(G, k) � h(G; k), and thus, gr(G; k) � h(G; k), by(35). Hence,G is graded-formal, by Lemma4.10. It follows thatm(G, k) � gr(G; k), and henceG is filtered-formal, by Lemma5.5.

Now supposeG filtered-formal. Then, by Lemma5.5, we have thatm(G, k) � gr(G, k).Thus, ifG is also graded-formal,m(G, k) � h(G; k). Hence,G is 1-formal. �

For instance, the Malcev Lie algebra of the free groupFn is the completion of the free gradedLie algebralie(kn); hence,Fn is 1-formal. In general, though, a filtered-formal group need notbe 1-formal. The examples include the free nilpotent groupsin Corollary7.3and the unipotentgroups in Example7.9. In fact, the triple Massey products in the cohomology of a filtered-formal group need not vanish (modulo indeterminacy).

Example 5.7. Let G = F2/Γ3F2 = 〈x1, x2 | [x1, [x1, x2]] = [x2, [x1, x2]] = 1〉 be the Heisenberggroup. ThenG is filtered-formal (see Corollary7.3 or Theorem7.5), yet has non-trivial tripleMassey products〈u1, u1, u2〉 and〈u2, u1, u2〉 in H2(G; k). Hence,G is not graded-formal.

As shown by in Hain in [25, 26] the Torelli groups in genus 4 or higher are 1-formal, but theTorelli group in genus 3 is filtered-formal, yet not graded-formal.

Proposition 5.8. Supposeφ : G1 → G2 is a homomorphism between two finitely-generatedgroups, inducing an isomorphism H1(G1; k) → H1(G2; k) and an epimorphism H2(G1; k) →H2(G2; k). Then we have the following statements.

(1) If G2 is 1-formal, then G1 is also1-formal.(2) If G2 is filtered-formal, then G1 is also filtered-formal.(3) If G2 is graded-formal, then G1 is also graded-formal.

Proof. A celebrated theorem of Stallings [65] (see also [18] and [22]) insures that the homo-morphismφ induces isomorphismsφk : (G1/ΓkG1) ⊗ k → (G2/ΓkG2) ⊗ k, for all k ≥ 1. Hence,φ induces an isomorphismm(φ) : m(G1; k)→ m(G2; k) between the respective Malcev comple-tions, thereby proving claim (1). Using now the isomorphism (35), the other two claims followat once. �

Corollary 5.9 ([54]). Let G be a finitely-generated group, with first Betti number either0 or 1.Then G is1-formal.


Proof. If b1(G) = 0, the homomorphismG → {1} induces an isomorphism onH1(−; k) and anepimorphism onH2(−; k). Likewise, if b1(G) = 1, the homomorphismG→ Gab→ Gab/tors�Z induces an isomorphism onH1(−; k) and an epimorphism onH2(−; k). Since free groups are1-formal, the claim follows from Proposition5.8. �

5.5. Split injections. We are now ready to state and prove the main result of this section, whichcompletes the proof of TheoremA from the Introduction.

Theorem 5.10. Let G be a finitely generated group. Suppose there is a split monomorphismι : K → G. The following statements then hold.

(1) If G is filtered-formal, then K is also filtered-formal.(2) If G is 1-formal, then K is also1-formal.

Proof. By hypothesis, we have an epimorphismσ : G։ K such thatσ ◦ ι = id. It follows thatthe induced mapsm(ι) and gr(ι) are also split injections.

Let π : F ։ G be a presentation forG. We then have an induced presentation forK, givenby the compositionπ1 := σπ : F ։ K. There is also a mapι1 : F → F which is a lift of ι,that is, ιπ1 = πι1. Consider the following diagram (for simplicity, we will suppress the zero-characteristic coefficient fieldk from the notation).

(37)

J1 lie(F) gr(K)

I1 lie(F) m(K)

J lie(F) gr(G)

I lie(F) m(G)

id

id Φ �

Φ1

m(ι1) m(ι)

gr(ι)

We havem(ι1) = gr(ι1). By assumption,G is filtered-formal; hence, there exists a filteredLie algebra isomorphismΦ : m(G) → gr(G) as in diagram (37), which induces the identity onassociated graded algebras. It follows thatΦ is induced from the identity map oflie(F) uponprojecting onto source and target, i.e., the bottom right square in the diagram commutes.

First, we show that the identity map id :lie(F) → lie(F) in the above diagram induces aninclusion mapI1 → J1. Suppose there is an elementc ∈ lie(F) such thatc ∈ I1 andc < J1, i.e.,[c] = 0 in m(K) and [c] , 0 in gr(G). Since gr(ι) is injective, we have that gr(ι)([c]) , 0, i.e.,gr(ι1)(c) < I . We also havem(ι)([c]) = 0 ∈ m(G), i.e.,m(ι1)(c) ∈ J. This contradicts the factthat the inclusionI → J is induced by the identity map. Thus,I1 ⊂ J1.

In view of the above, we may define a Lie algebra morphismΦ1 : m(K) → gr(K) as thequotient of the identity onlie(F). By construction,Φ1 is an epimorphism. We also have gr(ι) ◦Φ1 = Φ ◦ m(ι). Since the mapsm(ι), gr(ι) andΦ are all injective, the mapΦ1 is also injective.Therefore,Φ1 is an isomorphism, and so the groupK is filtered-formal.

Finally, part (2) follows at once from part (1) and Theorem4.11. �

This completes the proof of TheoremA from the Introduction. As we shall see in Example5.3, this theorem is useful for deciding whether certain infinite families of groups are 1-formal.


We now proceed with the proof of TheoremB. First, we need a lemma.

Lemma 5.11([17]). Let G1 and G2 be two finitely generated groups. Thenm(G1 × G2; k) �m(G1; k) ×m(G2; k) andm(G1 ∗G2; k) � m(G1; k) ∗m(G2; k).

Proposition 5.12. For any two finitely generated groups G1 and G2, the following conditionsare equivalent.

(1) G1 and G2 are filtered-formal.(2) G1 ∗G2 is filtered-formal.(3) G1 ×G2 is filtered-formal.

Proof. Since there exist split injections fromG1 andG2 to the productG1 ×G2 and coproductG1 ∗G2, we may apply Theorem5.10to conclude that implications (2)⇒(1) and (3)⇒(1) hold.Implications (1)⇒(2) and (1)⇒(3) follow from Lemmas2.7, 2.8, and5.11. �

Remark 5.13. As we shall see in Example11.15, the implication (1)⇒(3) from Proposi-tion 5.12cannot be strengthened from direct products to arbitrary semi-direct products. Moreprecisely, there exist split extensions of the formG = Fn ⋊α Z, for certain automorphismsα ∈ Aut(Fn), such that the groupG is not filtered-formal, although of course bothFn andZ are1-formal.

Corollary 5.14. Suppose G1 and G2 are finitely generated groups such that G1 is not graded-formal and G2 is not filtered-formal. Then the product G1×G2 and the free product G1 ∗G2 areneither graded-formal, nor filtered-formal.

Proof. Folows at once from Propositions4.15and5.12. �

As mentioned in the Introduction, concrete examples of groups which do not possess eitherformality property can be obtained by taking direct products of groups which enjoy one propertybut not the other.

6. Derived series and Lie algebras

We now study some of the relationships between the derived series of a group and the derivedseries of the corresponding Lie algebras.

6.1. Derived series. Consider the derived series of a groupG, starting atG(0) = G, G(1) = G′,andG(2) = G′′, and defined inductively byG(i+1) = [G(i),G(i)]. Note that any homomorphismφ : G → H takesG(i) to H(i). The quotient groups,G/G(i), are solvable; in particular,G/G′ =Gab, whileG/G′′ is the maximal metabelian quotient ofG.

AssumeG is a finitely generated group, and fix a coefficient fieldk of characteristic 0.

Proposition 6.1. The holonomy Lie algebras of the derived quotients of G are given by

(38) h(G/G(i); k) =

h(G; k)/h(G; k)′ for i = 1,

h(G; k) for i ≥ 2.

Proof. For i = 1, the statement trivially holds, so we may as well assumei ≥ 2. It is readilyproved by induction thatG(i) ⊆ Γ2i (G). Hence, the projections

(39) G // // G/G(i) // // G/Γ2iG

yield natural projectionsh(G; k) ։ h(G/G(i); k) ։ h(G/Γ2iG; k) = h(G; k). By Proposition4.8, the composition of these projections is an isomorphism of Lie algebras. Therefore, thesurjectionh(G; k)։ h(G/G(i); k) is an isomorphism. �


The next theorem is the Lie algebra version of Theorem 3.5 from [53].

Theorem 6.2([53]). For each i≥ 2, there is an isomorphism of complete, separated filteredLie algebras,

m(G/G(i)) � m(G)/m(G)(i),

wherem(G)(i) denotes the closure ofm(G)(i) with respect to the filtration topology onm(G).

6.2. Chen Lie algebras. As before, letG be a finitely generated group. For eachi ≥ 2, thei-thChen Lie algebraof G is defined to be the associated graded Lie algebra of the correspondingsolvable quotient, gr(G/G(i); k). Clearly, this construction is functorial.

The quotient map,qi : G ։ G/G(i), induces a surjective morphism between associatedgraded Lie algebras. Plainly, this morphism is the canonical identification in degree 1. Infact, more is true.

Lemma 6.3. For each i≥ 2, the mapgr(qi) : grk(G; k)։ grk(G/G(i); k) is an isomorphism for

each k≤ 2i − 1.

Proof. Taking associated graded Lie algebras in sequence (39) gives epimorphisms

(40) gr(G; k) // // gr(G/G(i); k) // // gr(G/Γ2i G; k) .

By a previous remark, the composition of these maps is an isomorphism in degreesk < 2i . Theconclusion follows. �

We now specialize to the case wheni = 2, which is the case originally studied by K.-T. Chenin [12]. TheChen ranksof G are defined asθk(G) := dimk(grk(G/G

′′; k)). For a free groupFn

of rankn, Chen showed that

(41) θk(Fn) = (k− 1)

(n+ k− 2

k

),

for all k ≥ 2. Let us also define theholonomy Chen ranksof G asθk(G) = dimk(h/h′′)k, whereh = h(G; k). It is readily seen thatθk(G) ≥ θk(G), with equality fork ≤ 2.

6.3. Chen Lie algebras and formality. We are now ready to state and prove the main resultof this section, which (together with the first corollary following it) proves TheoremC from theIntroduction.

Theorem 6.4. Let G be a finitely generated group. For each i≥ 2, the quotient map qi : G ։G/G(i) induces a natural epimorphism of gradedk-Lie algebras,

Ψ(i)G : gr(G; k)/ gr(G; k)(i) // // gr(G/G(i); k) .

Moreover, if G is a filtered-formal group, thenΨ(i)G is an isomorphism and the solvable quotient

G/G(i) is filtered-formal.

Proof. The mapqi : G ։ G/G(i) induces a natural epimorphism of gradedk-Lie algebras,gr(qi) : gr(G; k) ։ gr(G/G(i); k). By Proposition4.1, this epimorphism factors through anisomorphism, gr(G; k)/gr(G(i); k) ≃−→ gr(G/G(i); k), wheregr denotes the graded Lie algebraassociated with the filtrationΓkG(i) = ΓkG∩G(i).


On the other hand, as shown by Labute in [33, Proposition 10], the Lie ideal gr(G; k)(i) iscontained ingr(G(i); k). Therefore, the map gr(qi) factors through the claimed epimorphismΨ

(i)G , as indicated in the following commuting diagram,

(42) gr(G; k)

��

gr(qi )

)) ))❙❙❙❙❙

❙❙❙❙

❙❙❙❙

❙

gr(G; k)/ gr(G; k)(i)

��

Ψ(i)G // // gr(G/G(i); k) .

gr(G; k)/gr(G(i); k)

≃

55❧❧❧❧❧❧❧❧❧❧❧❧❧❧

Suppose now thatG is filtered-formal, and setm = m(G; k) andg = gr(G; k). We mayidentify g � m. Let g → g be the inclusion into the completion. Passing to solvable quotients,we obtain a morphism of filtered Lie algebras,

(43) ϕ(i) : g/g(i) // m/m(i) .

Passing to the associated graded Lie algebras, we obtain thefollowing diagram:

(44) g/g(i)

gr(ϕ(i))��

Ψ(i)G // gr(G/G(i); k)

�

��gr(m/m(i)) � // gr(m(G/G(i); k)).

All the graded Lie algebras in this diagram are generated in degree 1, and all the morphismsinduce the identity in this degree. Therefore, the diagram is commutative. Moreover, the rightvertical arrow from (43) is an isomorphism by Quillen’s isomorphism (35), while the lowerhorizontal arrow is an isomorphism by Theorem6.2.

Recall that, by assumption,m = g; therefore, the inclusion of filtered Lie algebrasg → ginduces a morphism between the following two exact sequences,

(45) 0 // gr(m(i)) // gr(m) // gr(m)/gr(m(i)) // 0

0 // g(i) //

OO

g //

�

OO

g/g(i) //

OO

0 .

Heregr means taking the associated graded Lie algebra corresponding to the induced filtration.Using formulas (2) and (9), it can be shown thatgr(m(i)) = g(i). Therefore, the morphismg/g(i) → gr(m)/gr(m(i)) is an isomorphism. We also know that gr(m/m(i)) = gr(m)/gr(m(i)).Hence, the map gr(ϕ(i)) is an isomorphism, and so, by (44), the mapΨ(i)

G is an isomorphism, too.By Lemma2.2, the mapϕ(i) induces an isomorphism of complete, filtered Lie algebras be-

tween the degree completion ofg/g(i) andm/m(i). As shown above,Ψ(i)G is an isomorphism;

hence, its degree completion is also an isomorphism. Composing with the isomorphism fromTheorem6.2, we obtain an isomorphism between the degree completiongr(G/G(i); k) and theMalcev Lie algebram(G/G(i); k). This shows that the solvable quotientG/G(i) is filtered-formal. �


Remark 6.5. In [33], Labute states that, in general, the morphismΨ(i)G from above is not an

isomorphism. It would be interesting to find an explicit example verifying this claim.

Returning now to the setup from Lemma4.6, let us compose the canonical projection gr(qi) :gr(G; k)։ gr(G/G(i); k) with the epimorphismΦG : h(G; k)։ gr(G; k). We obtain in this fash-ion an epimorphismh(G; k)։ gr(G/G(i); k), which fits into the following commuting diagram:

(46)

h(G) gr(G)

h(G/G(i)) gr(G/G(i))

h(G)/h(G)(i) gr(G)/ gr(G)(i) .

ΦG

Putting things together, we obtain the following corollary, which recasts Theorem 4.2 from[53] in a setting which is both functorial, and holds in wider generality. This corollary providesa way to detect non-1-formality of groups.

Corollary 6.6. For each for i≥ 2, there is a natural epimorphism of gradedk-Lie algebras,

Φ(i)G : h(G; k)/h(G; k)(i) // // gr(G/G(i); k) .

Moreover, if G is1-formal, thenΦ(i)G is an isomorphism.

Corollary 6.7. Suppose the group G is1-formal. Then, for each for i≥ 2, the solvable quotientG/G(i) is graded-formal if and only ifh(G; k)(i) vanishes.

Proof. By Proposition6.1, the canonical projectionqi : G → G/G(i) induces an isomorphismh(qi) : h(G; k)→ h(G/G(i); k). Since we assumeG is 1-formal, Corollary6.6guarantees that themapΦ(i)

G : h(G; k)/h(G; k)(i)→ gr(G/G(i); k) is an isomorphism. The claim follows from the leftsquare of diagram (46). �

7. Torsion-free nilpotent groups

In this section we study the graded and filtered formality properties of a well-known classof groups: that of finitely generated, torsion-free nilpotent groups. In the process, we proveTheoremD from the Introduction.

7.1. Nilpotent groups and Lie algebras. We start by reviewing the construction of the MalcevLie algebra of a finitely generated, torsion-free nilpotentgroupG (see [44, 35, 11] for moredetails). There is a refinement of the upper central series ofsuch a group,

(47) G = G1 > G2 > · · · > Gn > Gn+1 = 1,

with each subgroupGi < G a normal subgroup ofGi+1, and each quotientGi/Gi+1 an infinitecyclic group. (The integern is an invariant of the group, called the length ofG.) Using this fact,we can choose aMalcev basis{u1, . . . , un} for G, which satisfiesGi = 〈Gi+1, ui〉. Consequently,each elementu ∈ G can be written uniquely asua1

1 ua2

2 · · ·uann .

Using this basis, the groupG, as a set, can be identified withZn via the map sendingua11 · · ·u

ann

to a = (a1, . . . , an). The multiplication inG then takes the form

(48) ua11 · · ·u

ann · u

b11 · · ·u

bnn = uρ1(a,b)

1 · · ·uρn(a,b)n ,


whereρi : Zn × Zn → Z is a rational polynomial function, for each 1≤ i ≤ n. This procedureidentifies the groupG with the group (Zn, ρ), with multiplication the mapρ = (ρ1, . . . , ρn) : Zn×

Zn → Zn. Thus, we can define a simply-connected nilpotent Lie groupG ⊗ k = (kn, ρ) byextending the domain ofρ, which is called theMalcev completionof G.

The discrete groupG is a subgroup of the real Lie groupG ⊗ R. The quotient space,M =(G ⊗ R)/G, is a compact manifold, called anilmanifold. As shown by A. Malcev in [44], theLie algebra ofM is isomorphic tom(G;R). It is readily apparent that the nilmanifoldM isan Eilenberg–MacLane space of typeK(G, 1). As shown by Nomizu, the cohomology ringH∗(M,R) is isomorphic to the cohomology ring of the Lie algebram(G;R).

The polynomial functionsρi have the form

(49) ρi(a, b) = ai + bi + τi(a1, . . . , ai−1, b1, . . . , bi−1).

Denote byσ = (σ1, . . . , σn) the quadratic part ofρ. Thenkn can be given a Lie algebra structure,with bracket [a, b] = σ(a, b) − σ(b, a). As shown by Lambe and Priddy [35], this Lie algebra isisomorphic to the Malcev Lie algebram(G, k).

The group (Zn, ρ) has canonical basis{ei}ni=1, whereei is thei-th standard basis vector. Then

the Malcev Lie algebram(G; k) = (kn, [ , ]) has Lie bracket given by [ei , ej] =∑n

k=1 ski, jek, where

ski, j = bk(ei , ej) − bk(ej , ei).

Remark 7.1. Let m = m(G; k) be the Malcev Lie algebra from above. Note that gr(m) = kn

has the same basise1, . . . , en asm, but, in general, the Lie bracket on gr(m) is different. TheLie algebram (and thus, the groupG) is filtered-formal if and only ifm � gr(m), as filtered Liealgebras. In general, though, this isomorphism need not preserve the chosen basis.

7.2. Nilpotent quotients and filtered formality. Let G be a finitely generated group. IfG isnilpotent, then clearlygr(G; k) = gr(G; k). On the other hand, ifG is filtered-formal, then bydefinitionm(G; k) � gr(G; k). Thus, ifG is both nilpotent and filtered-formal, then the MalcevLie algebram(G; k) admits the structure of a graded Lie algebra.

The next theorem shows that filtered-formality is inheritedupon taking nilpotent quotients.

Theorem 7.2. Let G be a finitely generated group, and suppose G is filtered-formal. Then allthe nilpotent quotients G/Γi(G) are filtered-formal.

Proof. Set g = gr(G; k) andm = m(G; k), and writeg =⊕

k≥1 gk. Then, for eachi ≥ 1,the canonical projectionφi : G ։ G/ΓiG induces an epimorphism of complete, filtered Liealgebras,m(φi) : m։ m(G/ΓiG; k). In each degreek ≥ i, we have thatΓkm(G/ΓiG; k) = 0, andsom(φi)(Γkm) = 0. Therefore, there exists an induced epimorphism

(50) Φk,i : m/Γkm // // m(G/ΓiG; k) .

Passing to associated graded Lie algebras, we obtain an epimorphism gr(Φk,i) : gr(m/Γkm)։gr(m(G/ΓiG; k)), which is readily seen to be an isomorphism fork = i. Using now Lemma2.2,we conclude that the mapΦi,i is an isomorphism of completed, filtered Lie algebras.

On the other hand, our filtered-formality assumption onG allows us to identifym � g =∏k≥1 gk. Using now formula (7), we find thatm/Γkm = g/Γkg = g/Γkg, for all k ≥ 1. Us-

ing these identifications fork = i, together with the isomorphismΦi,i from above, we obtainisomorphisms

(51) m(G/ΓiG; k) � g/Γig � gr(G/ΓiG; k).

This shows that the nilpotent quotientG/ΓiG is filtered-formal, and we are done. �


Corollary 7.3. Let F be a finitely generated free group. Then the n-step, freenilpotent groupF/Γn+1F is filtered-formal.

Proof. The free groupF is 1-formal, and thus filtered-formal. By Theorem7.2, then, eachnilpotent quotient ofF is also filtered-formal. �

Remark 7.4. Alternatively, we could use [48, Corollary 2.14], which says thatm(F/ΓkF) �L/(ΓkL ), whereL = lie(F) is the free Lie algebra associated toF. Since gr(F/ΓkF) � L/(ΓkL ),this gives another proof thatF/ΓkF is filtered-formal, for allk ≥ 2.

7.3. Torsion-free nilpotent groups and filtered formality. We now study in more detail thefiltered-formality properties of torsion-free nilpotent groups. We start by singling out a ratherlarge class of groups which enjoy this property.

Theorem 7.5. Let G be a finitely generated, torsion-free,2-step nilpotent group. If Gab istorsion-free, then G is filtered-formal.

Proof. The lower central series of our group takes the formG = Γ1G > Γ2G > Γ3G = 1. Let{x1, . . . , xn} be a basis forG/Γ2G = Zn, and let{y1, . . . , ym} be a basis forΓ2G = Zm. Then, asshown for instance in [28, Lemma 6.1], the groupG has presentation

(52) G =⟨x1, . . . , xn, y1, . . . , ym

∣∣∣∣ [xi , x j ] =m∏

k=1

yck

i, j

k , [yi , y j] = 1, for i < j; [xi , y j] = 1⟩.

Let a, b ∈ Zn+m. A routine computation shows that

ρi(a, b) = ai + bi , for 1 ≤ i ≤ n,(53)

ρn+k(a, b) = an+k + bn+k −

k∑

j=1

n∑

i= j+1

ckj,iaib j, for 1 ≤ k ≤ m.

Setckj,i = −ck

i, j if j > i. It follows that the Malcev Lie algebram(G; k) = (kn+m, [ , ]) has Lie

bracket given on generators by [ei , ej] =∑m

k=1 cki, jen+k for 1 ≤ i , j ≤ n, and zero otherwise.

Turning now to the associated graded Lie algebra of our group, we have an additive decom-position, gr(G; k) = gr1(G; k) ⊕ gr2(G; k) = kn ⊕ km, where the first factor has basis{e1, . . . , en},the second factor has basis{en+1, . . . , en+m}, and the Lie bracket is given as above. Therefore,m(G) � gr(G; k), as filtered Lie algebras. Hence,G is filtered-formal. �

Furthermore, as shown by Cornulier [15], all nilpotent Lie algebras of dimension 4 or lessare filtered-formal. In general, though, finitely-generated, torsion-free nilpotent groups neednot be filtered-formal. We illustrate this phenomenon with two examples: the first one extractedfrom [15], and the second one adapted from Lambe and Priddy [35].

Example 7.6. Let m be the 5-dimensionalk-Lie algebra with non-zero Lie brackets given by[e1, e3] = e4, [e1, e4] = e5, and [e2, e3] = e5. It is readily checked that the center ofm is 1-dimensional, generated bye5, while the center of gr(m) is 2-dimensional, generated bye4 ande5. Therefore,m � gr(m), and som is not filtered-formal.

Example 7.7. Let m be the 5-dimensionalk-Lie algebra with non-zero Lie brackets given by[e2, e3] = e6, [e2, e4] = e7, [e2, e5] = −e7, [e3, e4] = e7, and [e1, ei ] = ei+1 for 2 ≤ i ≤ 6. Weclaim thatm is not isomorphic to gr(m), as Lie algebras.


Indeed, supposeφ : m → gr(m) is an isomorphism of the underlying vector spaces, pre-serving Lie brackets. Choose a basis{z1, . . . , z7} for gr(m) = k7. Thenφ is given by a ma-trix A = (ai j ) such thatai j = 0 for 3 ≤ i ≤ 7 and i > j ≥ 1. Since [φ(e2), φ(e3)] =a21a33z3 + a21a34z4 + a21a35z6 + a21a36z7 andφ(e6) = a66z6 + a67z7, we must havea21a33 = 0anda21a35 = a66. Moreover, since det(A) , 0, we must also haveaii , 0 for i ≥ 3. But this isimpossible, and the claim is proved.

In both examples, the nilpotent Lie algebram in question may be realized as the Malcev Liealgebram(G; k) of a finitely-generated, torsion-free nilpotent groupG. Sincem is not filtered-formal, the groupG is not filtered-formal, either.

7.4. Graded formality and Koszulness. Carlson and Toledo [9] classified finitely generated,1-formal, nilpotent groups with first Betti number 5 or less,while Plantiko [57] gave sufficientconditions for the associated graded Lie algebras of such groups to be non-quadratic. Thefollowing proposition follows from Theorem 4.1 in [57] and Lemma 2.4 in [9].

Proposition 7.8 ([9, 57]). Let G = F/R be a finitely presented, torsion-free, nilpotent group.If there exists a decomposable element in the kernel of the cup product H1(G; k) ∧ H1(G; k) →H2(G; k), then G is not graded-formal.

Example 7.9. Let Un(R) be the nilpotent Lie group of upper triangular matrices with 1’s alongthe diagonal. The quotientM = Un(R)/Un(Z) is a nilmanifold of dimensionN = n(n− 1)/2.The unipotent groupUn(Z) has canonical basis{ui j | 1 ≤ i < j ≤ n}, whereui j is the matrixobtained from the identity matrix by putting 1 in position (i, j). Moreover,Un(Z) � (ZN, ρ),whereρi j (a, b) = ai j +bi j +

∑i<k< j aikbk j, see [35]. The unipotent groupUn(Z) is filtered-formal;

nevertheless, Proposition7.8shows that this group is not graded-formal forn ≥ 3.

Proposition 7.10. Let G be a finitely generated, torsion-free, nilpotent group, and suppose G isfiltered-formal. Then G is abelian if and only if the algebra U(gr(G; k)) is Koszul.

Proof. We only need to deal with the proof of the non-trivial direction. If the algebraU =U(gr(G; k)) is Koszul, then the Lie algebra gr(G; k) is quadratic, i.e., the groupG is graded-formal. Under the assumption thatG is filtered-formal, we then have thatG is 1-formal.

Let M be the nilmanifold with fundamental groupG. ThenM is also 1-formal. By Nomizu’stheorem, the cohomology ringA = H∗(M; k) is isomorphic to the Yoneda algebra Ext∗

U(k, k).On the other hand, sinceU is Koszul, the Yoneda algebra is isomorphic toU !, which is alsoKoszul. Hence,A is a Koszul algebra. As shown by Papadima and Yuzvinsky [56], if M is1-formal and ifA is Koszul, thenM is formal. On the other, as shown by Hasegawa [27], M isformal if and only ifM is a torus. This completes the proof. �

Corollary 7.11. Let G be a finitely generated, torsion-free,2-step nilpotent group. If Gab istorsion-free, then U(gr(G; k)) is not Koszul.

Example 7.12. Let G = 〈x1, x2, x3, x4 | [x1, x3], [x1, x4], [x2, x3], [x2, x4], [x1, x2][ x3, x4]〉. ThegroupG is a 2-step, commutator-relators nilpotent group. Hence, by the above corollary, theenveloping algebraU(h(G; k)) is not Koszul. In fact,U(h(G; k))! is isomorphic to the quadraticalgebra from Example3.9, which is not Koszul.

8. Magnus expansions for finitely generated groups

The Magnus expansion for free groups involves the Fox free derivatives, which can be usedto compute the cup products and the Massey products in the cohomology ring of a commutator-relators group (see [21, 50]). In this section, we introduce and study a Magnus-type expansion


for arbitrary finitely generated groups. Such an expansion will be used later on to compute cupproducts and find an explicit presentation for the holonomy Lie algebra.

8.1. The Magnus expansion for a free group.As before, letk denote a field of character-istic 0. Let F be the free group generated byx = {x1, . . . , xn}, and setFk = Fab ⊗ k. Thecompleted tensor algebraT(Fk) can be identified withk〈〈x〉〉, the power series ring overk in nnon-commuting variables.

Let kF be the group ring ofF, with augmentation mapǫ : kF → k given byǫ(xi) = 1. Thereis a well-defined ring morphismM : kF → k〈〈x〉〉, called theMagnus expansion, given by

(54) M(xi) = 1+ xi and M(x−1i ) = 1− xi + x2

i − x3i + · · · .

The Fox derivativesare the ring morphisms∂i : ZF → ZF defined by the rules∂i(1) = 0,∂i(x j) = δi j , and∂i(uv) = ∂i(u)ǫ(v) + u∂i(v) for u, v ∈ ZF. The higher Fox derivatives∂i1,...,ik arethen defined inductively.

The Magnus expansion can be computed in terms of Fox derivatives, as follows. Giveny ∈ F, if we write M(y) = 1+

∑aI xI , thenaI = ǫI (y), whereI = (i1, . . . , is), andǫI = ǫ ◦ ∂I is

the composition ofǫ : kF → k with ∂I : kF → kF. Let Mk be the composite

(55) kFM //

Mk

))T(Fk)

grk // grk(T(Fk)) ,

In particular, for eachy ∈ F, we haveM1(y) =∑n

i=1 ǫi(y)xi, while for eachy ∈ [F, F] we have

(56) M2(y) =∑

i< j

ǫi, j(y)(xi x j − x j xi).

Notice thatM2(y) is a primitive element in the Hopf algebraT(Fk), which corresponds to theelement

∑i< j ǫi, j(y)[xi, x j ] in the free Lie algebralie(Fk).

Remark 8.1. The mapM extends to a mapM : kF → T(Fk) which is an isomorphism ofcomplete, filtered algebras, butM is not compatible with the respective comultiplications ifrankF > 1. On the other hand, Lin constructed in [40] an exponential expansion, exp :kF →T(Fk), while Massuyeau showed in [48] that the map exp is an isomorphism of complete Hopfalgebras. Restricting this map to the Lie algebras of primitive elements gives an isomorphismm(G, k) ≃−→ lie(Fk).

8.2. The Magnus expansion for finitely generated groups.Given a finitely generated groupG, there exists an epimorphismϕ : F ։ G from a free groupF of finite rank. Letπ be theinduced epimorphism in homology fromFk := H1(F; k) to Gk := H1(G; k).

Definition 8.2. The Magnus expansionfor a finitely generated groupG, denoted byκ, is thecomposite

(57) kFM //

κ

((T(Fk)

T(π) // T(Gk) ,

whereM is the classical Magnus expansion for the free groupF, and the morphismT(π) : T(Fk)։T(Gk) is induced by the projectionπ : Fk → Gk.


In particular, if the groupG is a commutator-relators group, thenπ identifiesGk with Fk, andthe Magnus expansionκ coincides with the classical Magnus expansionM.

More generally, letG be a group generated byx = {x1, . . . , xn}, and letF be the free groupgenerated by the same set. Pick a basisy = {y1, . . . , yb} for Gk, and identifyT(Gk) with k〈〈y〉〉.Let κ(r)I be the coefficient ofyI := yi1 · · · yis in κ(r), for I = (i1, . . . , is). Then we can write

(58) κ(r) = 1+∑

I

κ(r)I · yI .

Lemma 8.3. If r ∈ ΓkF, thenκ(r)I = 0, for |I | < k. Furthermore, if r∈ Γ2F, thenκ(r)i, j =

−κ(r) j,i.

Proof. SinceM(r)I = ǫI (r) = 0 for |I | < k (see for instance [50]), we have thatκ(r)I = 0 for|I | < k. To prove the second assertion, identify the completed symmetric algebrasSym(Fk) andSym(Gk) with the power series ringsk[[x]] and k[[y]] in the following commutative diagram oflinear maps.

(59) kF

κ

!!❈❈❈❈❈❈❈❈❈

M // T(Fk)

T(π)��

α1 // Sym(Fk)

Sym(π)��

T(Gk)α2 // Sym(Gk).

Whenr ∈ [F, F], we have thatα2 ◦ κ(r) = Sym(π) ◦ α1 ◦ M(r) = 1. Thus,κi(r) = 0 andκ(r)i, j + κ(r) j,i = 0. �

Lemma 8.4. If u, v ∈ F satisfyκ(u)J = κ(v)J = 0 for all |J| < s, for some s≥ 2, then

κ(uv)I = κ(u)I + κ(v)I , for |I | = s.

Moreover, the above formula is always true for s= 1.

Proof. We have thatκ(uv) = κ(u)κ(v) for u, v ∈ F. If κ(u)J = κ(v)J = 0 for all |J| < s, thenκ(u) = 1+

∑|I |=s κ(u)I yI up to higher-order terms, and similarly forκ(v). Then

(60) κ(uv) = κ(u)κ(v) = 1+∑

|I |=s

(κ(u)I + κ(u)I )yI + higher-order terms.

Therefore,κ(uv)i = κ(u)i + κ(v)i , and soκ(uv)I = κ(u)I + κ(v)I . �

8.3. Truncating the Magnus expansions.Recall from (55) that we defined truncationsMk

of the Magnus expansionM of a free groupF. In a similar manner, we can also define thetruncations of the Magnus expansionκ for any finitely generated groupG.

Lemma 8.5. The following diagram commutes.

(61) kF

κ

!!❇❇❇❇❇❇❇❇❇

M // T(Fk)

T(π)��

grk // grk(T(Fk)) =⊗kkn

⊗kπ

��

T(Gk)grk // grk(T(Gk)) =

⊗kkb

Proof. The triangle on the left of diagram (61) commutes, since it consists of ring homomor-phisms by the definition of the Magnus expansion for a group.

The morphisms in the square on the right side of (61) are homomorphisms betweenk-vectorspaces. The square commutes, sinceπ is a linear map. �


In diagram (61), denote the composition ofκ and grk by κk.

(62) kFκ //

κk

))T(Gk)

grk // grk(T(Gk)) ,

In particular,κ1(r) =∑b

i=1 κ(r)iyi for r ∈ F. By Lemma8.3, if r ∈ [F, F], then

(63) κ2(r) =∑

1≤i< j≤b

κ(r)i, j(yiy j − y jyi).

The next lemma provides a close connection between the Magnus expansionκ and the clas-sical Magnus expansionM.

Lemma 8.6. Let (ai,s) be the b× n matrix for the linear mapπ : Fk → Gk, and let r∈ F be anarbitrary element. Then, for each1 ≤ i, j ≤ b, we have

κ(r)i =

n∑

s=1

ai,sǫs(r),(64)

κ(r)i, j =

n∑

s,t=1

(ai,sa j,tǫs,t(r)

).(65)

Proof. By assumption, we haveπ(xs) =∑b

i=1 ai,syi . By Lemma8.5(for k = 1), we have

κ1(r) = π ◦ M1(r) = π(n∑

s=1

ǫs(r)xs) =n∑

s=1

b∑

i=1

ai,sǫs(r)yi ,

which gives formula (64). By Lemma8.5(for k = 2), we have

κ2(r) = π ⊗ π ◦ M2(r) = π ⊗ π

n∑

s,t=1

ǫs,t(r)xs ⊗ xt

=n∑

s,t=1

b∑

i, j=1

ǫs,t(r)ai,sa j,tyi ⊗ y j ,

which gives formula (65). �

8.4. Echelon presentations.Let G be a group with finite presentationP = F/R = 〈x | w〉wherex = {x1, . . . , xn} andw = {w1, . . . ,wm}. ThenR is a free subgroup ofF generated by thesetw, andRab is a free abelian group with the same generating set.

Let KP be the 2-complex associated to this presentation ofG. We may viewx as a basis forC1(KP; k) andw as a basis forC2(KP; k) = km. With this choice of bases, the matrix of theboundary mapdP

2 : C2(KP; k)→ C1(KP; k) is them× n Jacobian matrixJP = (ǫi(wk)).

Definition 8.7. A groupG has anechelon presentation P= 〈x | w〉 if the matrix (ǫi(wk)) is inrow-echelon form.

Example 8.8. Let G be the group generated byx1, . . . , x6, with relationsw1 = x21x1

2x33x5

4,w2 = x2

3x−24 x4

6, w3 = x34x−2

5 x36, andw4 = [x1, x2]. The given presentation is already an eche-

lon presentation, since the matrixdG2 =

(2 1 3 5 0 00 0 2 −2 0 40 0 0 3 −2 30 0 0 0 0 0

)has the required form.

The next proposition shows that for any finitely generated group, we can construct a groupwith an echelon presentation such that the two groups have the same holonomy Lie algebra.


Proposition 8.9. Let G be a group with a finite presentation P. There exists thena groupG withan echelon presentationP and a surjective homomorphismρ : G ։ G inducing the followingisomorphisms:

(i) ρ∗ : Hi(KP; k) ≃−→ Hi(KP; k) for i = 1, 2;(ii) ρ∗ : H i(KP; k) ≃−→ H i(KP; k) for i = 1, 2;(iii) h(ρ) : h(G; k) ≃−→ h(G; k).

Proof. SupposeG has presentationP = 〈x | r 〉, wherex = {x1, . . . , xn} andr = {r1, . . . , rm}.Consider the diagram

(66) H2(KG; k)

�

��

// C2(KG; k)

�

��

dG2 // C1(KG; k)

�

��

π // Gk

�

��H2(KG; k) C2(KG; k)oo C1(KG; k)

(dG2 )∗

oo H1(KG; k),π∗oo

where the vertical arrows indicate duality isomorphisms. By Gaussian elimination overZ, thereexists a matrixC = (cl,k) ∈ GL(m;Z) such thatC · (dG

2 )∗ is in row-echelon form. We define anew group,

(67) G = 〈x1, . . . , xn | w1, . . . ,wm〉,

wherewk = rc1,k

1 rc2,k

2 · · · rcm,km for 1 ≤ k ≤ m. The surjectionρ : G ։ G, defined by sending a

generatorxi ∈ G to the same generatorxi ∈ G for 1 ≤ i ≤ n, induces a chain map from thecellular chain complexC∗(KG; k) to C∗(KG; k), as follows

(68) C0(KG; k) C1(KG; k)dG

1 =0oo C2(KG; k)

dG2oo 0oo · · ·oo

C0(KG; k)

ρ0=id

OO

C1(KG; k)dG

1 =0oo

ρ1=id

OO

C2(KG; k)dG

2oo

ρ2

OO

0oo

OO

· · ·oo

The mapρ2 is given by the matrixC, while dG2 is given by the compositiondG

2 ◦ ρ2. Thehomomorphismρ induces isomorphisms on homology groups. In particular,ρ∗ : H1(KG; k) →H1(KG; k) is the identity. Then, we see thatπG = πG.

The last statement follows from the functoriality of the cup-product and Lemma4.4. �

9. Group presentations and (co)homology

We compute in this section the cup products of degree 1 classes in the cohomology of afinitely presented group in terms of the Magnus expansion associated to the group.

9.1. A chain transformation. We start by reviewing the classical bar construction. LetG be adiscrete group, and letB∗(G) be the normalized bar resolution (see [21, 7]), whereBp(G) is thefree leftZG-module on generators [g1| . . . |gp], with gi ∈ G andgi , 1, andB0(G) = ZG is freeon one generator, [ ]. The boundary operators areG-module homomorphisms,dp : Bp(G) →Bp−1(G), defined by

(69) dp[g1| . . . |gp] = g1[g2| . . . |gp] +p−1∑

i=1

(−1)i[g1| . . . |gigi+1| . . . |gp] + (−1)p[g1| . . . |gp−1].


In particular,d1[g] = (g− 1)[ ] andd2[g1|g2] = g1[g2] − [g1g2] + [g1]. Let ǫ : B0(G)→ k be theaugmentation map. We then have a free resolution of the trivial G-modulek,

(70) · · · // B2(G)d2 // B1(G)

d1 // B0(G)ǫ // k // 0 .

As before,k will denote a field of characteristic 0. We shall viewk as a rightkG-module,with action induced by the augmentation map. An element of the cochain groupBp(G; k) =HomkG(Bp(G), k) can be viewed as a set functionu: Gp → k satisfying the normalizationconditionu(g1, . . . , gp) = 0 if somegi = 1. The cup-product of two 1-dimensional classesu, u′ ∈ H1(G; k) � B1(G; k) � Hom(G, k) is given by

(71) u∪ u′[g1|g2] = u(g1)u′(g2).

Now suppose the groupG = 〈x1, . . . xn | r1, . . . , rm〉 is finitely presented. Letϕ : F ։ Gbe the presenting homomorphism, and letKG be the 2-complex associated to this presentationof G. Denote the cellular chain complex (overk) of the universal cover of this 2-complex byC∗(KG). The differentials in this chain complex are given by

δ1(λ1, . . . , λn) =n∑

i=1

λi(xi − 1),(72)

δ2(µ1, . . . , µm) =( m∑

j=1

µ jϕ(∂1w j), . . . ,m∑

j=1

µ jϕ(∂nw j)),

for λi , µ j ∈ ZG.

Lemma 9.1([21]). There exists a chain transformation T: C∗(KG) → B∗(G) commuting withthe augmentation map,

0 Zoo C0(KG)ǫoo

T0

��

C1(KG)δ1oo

T1

��

C2(KG)δ2oo

T2

��

0oo

��

· · ·oo

0 Zoo B0(G)ǫoo B1(G)

d1oo B2(G)d2oo B3(G)oo · · ·oo

Here

(73) T0(λ) := λ[ ] , T1(λ1, . . . , λn) =∑

i

λi [xi ], T2(µ1, . . . , µm) =m∑

j=1

µ jτ1T1δ2(ej),

where ej = (0, . . . , 0, 1, 0, . . . , 0) ∈ (ZG)m has a1 only in position j, andτ0 : B0(G) → B1(G)andτ1 : B1(G)→ B2(G) are the homomorphisms defined by

(74) τ0(g[ ]) = [g] andτ1(g[g1]) = [g|g1],

for all g, g1 ∈ G.

9.2. Cup products for echelon presentations.Now letG be a group with echelon presentationG = 〈x | w〉, wherex = {x1, . . . , xn} andw = {w1, . . . ,wm}, as in Definition8.7. Suppose thepivot elements of them× n matrix (ǫi(wk)) are in position{i1, . . . , id}, and letb = n− d.

Lemma 9.2. Let KG be the2-complex associated to the above presentation for G. Then

(i) The vector space H1(KG; k) = kb has basisy = {y1, . . . , yb}, where yj = xid+ j for1 ≤ j ≤ b.


(ii) The vector space H2(KG; k) = km−d has basis{wd+1, . . . ,wm} which coincide with theaugmentation of the basis{ed+1, . . . , em} in Lemma9.1.

Proof. The lemma follows from the fact that the matrix (ǫi(wk)) is in row-echelon form. �

We will choose as basis forH1(KG; k) the set{u1, . . . , ub}, whereui is the Kronecker dual toyi .

Lemma 9.3. For each basis element ui ∈ H1(KG; k) � H1(G; k) as above, and each r∈ F, wehave that

ui([ϕ(r)]) =n∑

s=1

ǫs(r)ai,s = κi(r),

where(ai,s)b×n is the matrix for the projection mapπ : Fk → Gk.

Proof. If r ∈ F, thenϕ(r) ∈ G and [ϕ(r)] ∈ B1(G). Hence,

(75) ui([ϕ(r)]) =n∑

s=1

ǫs(r)ui([xs]) =n∑

s=1

ǫs(r)ai,s = κi(r).

SinceH1(G; k) � B1(G; k) � Hom(G, k), we may viewui as a group homomorphism. Thisyields the first equality. Sinceπ(xs) =

∑bj=1 ai,syi andui = y∗i , the second equality follows. The

last equality follows from Lemma8.6. �

Theorem 9.4. The cup-product map H1(KG; k) ∧ H1(KG; k)→ H2(KG; k) is given by

(ui ∪ u j ,wk) = κ(wk)i, j ,

for 1 ≤ i, j ≤ b and d+ 1 ≤ k ≤ m, whereκ is the Magnus expansion of G.

Proof. Let us write the Fox derivative∂t(wk) as a finite sum,∑

x∈F pxtkx, for 1 ≤ t ≤ n, and

1 ≤ k ≤ m. We then have

T2(ek) = τ1T1(δ2(ek)) by (73)

= τ1T1 (ϕ(∂1(wk)), . . . , ϕ(∂n(wk))) by (72)(76)

= τ1

( n∑

t=1

ϕ(∂t(wk)

)[xt]

)by (73)

=

n∑

t=1

∑

x∈F

pxtk[ϕ(x)|xt]. by (74)


The chain transformationT : C∗(KG)→ B∗(G) induces an isomorphism on first cohomology,T∗ : H1(G; k)→ H1(KG; k). Let us viewui andu j as elements inH1(G; k). We then have

(ui ∪ u j , 1⊗ZG ek) = (ui ∪ u j , 1⊗ZG T2(ek))

= (ui ∪ u j ,

n∑

t=1

∑

x∈F

pxtk[ϕ(x)|xt]) by (76)

=

n∑

t=1

∑

x∈F

pxtkui(ϕ(x))u j(xt) by (71)

=

n∑

t=1

∑

x∈F

pxtkui(ϕ(x))a j,t by Lemma9.3

=

n∑

t=1

∑

x∈F

pxtk

n∑

s=1

ai,sǫs(x)a j,t by Lemma9.3

=

n∑

t=1

n∑

s=1

(a j,tai,sǫs,t(wk)

)Fox derivative

= κ(wk)i, j , by Lemma8.6

and this completes the proof. �

Example 9.5. Let KG be the presentation 2-complex for the groupG in Example8.8, with thehomology basis as in Lemma9.2. A basis ofH1(KG; k) = k3 is {x2, x5, x6} while a basis ofH2(KG; k) = k is {w4}, and a basis ofH1(KG; k) is {u1, u2, u3}. With these choices, we have that(u1 ∪ u2,w4) = 8/3, (u1 ∪ u3,w4) = −7, and (u2 ∪ u3,w4) = 0.

9.3. Cup products for finite presentations. LetG be a group with a finite presentation〈x | r 〉.By Proposition8.9, there exists a groupG with an echelon presentation〈x | w〉.

Let us choose a basisy = {y1, . . . , yb} for H1(KG; k) � H1(KG; k) and the dual basis{u1, . . . , ub}

for H1(KG; k) � H1(KG; k). Choose also a basis{r1, . . . , rm} for C2(KG; k) and a basis{w1, . . . ,wm}

for C2(KG; k). Set

(77) γk := ρ∗(wk) =m∑

l=1

cl,kr l

Then {γk | 1 ≤ k ≤ m} is another basis forC2(KG; k). Furthermore,{wd+1, . . . ,wm} is a ba-sis for H2(KG; k) and{γd+1, . . . , γm} is a basis forH2(KG; k). Thus,H2(KG; k) has dual basis{βd+1, . . . , βm}.

Theorem 9.6. The cup-product map∪ : H1(KG; k) ∧ H1(KG; k) → H2(KG; k) is given by theformula

ui ∪ u j =

m∑

k=d+1

κ(wk)i, jβk,

That is,(ui ∪ u j , γk) = κ(wk)i, j for all 1 ≤ i, j ≤ b.


Proof. By Proposition8.9, we have thatγk := ρ∗(wk) = ρ∗(1 ⊗ZG ek), for all d + 1 ≤ k ≤ m.Hence,

(ui ∪ u j, γk) =(ui ∪ u j , ρ∗(1⊗ZG ek)

)

=(ρ∗(ui ∪ u j), 1⊗ZG ek

)

=(ui ∪ u j , 1⊗ZG ek

)sinceρ∗(ui) = ui

= κ(wk)i, j by Theorem9.4.

The claim follows. �

Corollary 9.7 ([21]). For a commutator-relators group G= 〈x | r 〉, the cup-product mapH1(KG; k) ∧ H1(KG; k)→ H2(KG; k) is given by

(ui ∪ u j, rk) = M(rk)i, j ,

for 1 ≤ i, j ≤ n and1 ≤ k ≤ m.

Remark 9.8. In [21], Fenn and Sjerve also gave formulas for the higher-order Massey productsin a commutator-relator group, using the classical Magnus expansion. For instance, supposeG = 〈x | r〉, where the single relatorr belongs to [F, F] and is not a proper power. LetI =(i1, . . . , ik), and supposeǫis,...,it−1(r) = 0 for all 1 ≤ s < t ≤ k + 1, (s, t) , (1, k + 1). Then theevaluation of the Massey product〈−ui1, . . . ,−uik〉 on the homology class [r] ∈ H2(G;Z) equalsǫI (r). For an alternative approach, in a more general context, see [59, Theorem 2].

10. A presentation for the holonomy Lie algebra

In this section, we give a presentation for the holonomy Lie algebra and the Chen holonomyLie algebra of a finitely presented group. In the process, we complete the proof of the first twoparts of TheoremE from the Introduction.

10.1. Magnus expansion and holonomy.In view of Theorem8.9, for any group with finitepresentation〈x | r 〉, there exists a group with echelon presentationP = 〈x | w〉 such that the twogroups have the same holonomy Lie algebras.

Let G = F/Rbe a group admitting an echelon presentationP as above, withx = {x1, . . . , xn}

andw = {w1, . . . ,wm}. We now give a more explicit presentation for the holonomy Lie algebrah(G; k) overk.

Let ∂i(wk) ∈ ZF be the Fox derivatives of the relations, and letǫi(wk) ∈ Z be their augmen-tations. Recall from Lemma9.2that we can choose a basisy = {y1, . . . , yb} for H1(KP; k) and abasis{wd+1, . . . ,wm} for H2(KP; k), whered is the rank of Jacobian matrixJP = (ǫi(wk)), viewedas anm× n matrix overk. Let lie(y) be the free Lie algebra overk generated byy in degree 1.Recall thatκ2 is the degree 2 part of the Magnus expansion ofG given explicitly in (63). Thus,we can identifyκ2(wk) with

∑i< j κ(wk)i, j [yi, y j ] in lie(y) for d+ 1 ≤ k ≤ m.

Theorem 10.1. Let G be a group admitting an echelon presentation G= 〈x | w〉. Then thereexists an isomorphism of graded Lie algebras

h(G; k)�

−−→ lie(y)/ideal(κ2(wd+1), . . . , κ2(wm)) .

Proof. Combining Theorem9.4 with the fact that (ui ∧ u j,∪∗(wk)) = (∪(ui ∧ u j),wk), we see

that the dual cup-product map,∪∗ : H2(KP; k)→ H1(KP; k) ∧ H1(KP; k), is given by

(78) ∪∗ (wk) =∑

1≤i< j≤b

κ(wk)i, j(yi ∧ y j).


Hence, the following diagram commutes.

(79) H2(KP; k) ∪∗ //� _

��

H1(KP; k) ∧ H1(KP; k)� _

��C2(KP; k)

κ2 // H1(KP; k) ⊗ H1(KP; k)

Using now the identification ofκ2(wk) and∑

i< j κ(wk)i, j[yi , y j ] as elements oflie(y), the defi-nition of the holonomy Lie algebra, and the fact thath(G; k) � h(KP; k), we arrive at the desiredconclusion. �

Corollary 10.2. The universal enveloping algebra U(h) of h(G; k) has presentation

U(h) = k〈y〉/ ideal(κ2(wn−b+1), . . . , κ2(wm)).

Recall that, forr ∈ [F, F], the primitive elementM2(r) in T2(Fk) corresponds to the element∑i< j ǫi, j(r)[xi, x j ] in lie2(x).

Corollary 10.3 ([53]). If G = 〈x | r 〉 is a commutator-relators group, then

h(G; k) = lie(x)/ideal{∑

i< j

ǫi, j(r)[xi, x j ] | r ∈ r}.

Corollary 10.4. For every quadratic, rationally defined Lie algebrag, there exists a commutator-relators group G such thath(G; k) � g.

Proof. By assumption, we may writeg = lie(x)/a, wherea is an ideal generated by elements ofthe formℓk =

∑ci jk [xi , x j ] for 1 ≤ k ≤ m, and where the coefficientsci jk are inQ. Clearing

denominators, we may assume allci jk are integers. We can then define wordsrk = [xi , x j ]ci jk inthe free group generated byx, and setG = 〈x | r1, . . . , rm〉. The desired conclusion follows fromCorollary10.3. �

10.2. Presentations for the holonomy Chen Lie algebras.The next result (which completesthe proof of TheoremE from the Introduction) sharpens and extends the first part ofTheorem7.3 from [53].

Theorem 10.5. Let G = 〈x | r 〉 be a finitely presented group, and seth = h(G, k). Let y ={y1, . . . , yb} be a basis of H1(G; k). Then, for each i≥ 2,

h/h(i) � lie(y)/(ideal(κ2(wn−b+1), . . . , κ2(wm)) + lie(i)(y)),

where b= b1(G) and wk is defined in(67).

Proof. By Theorem10.1, the holonomy Lie algebrah is isomorphic to the quotient of the freeLie algebralie(y) by the ideal generated byκ2(wn−b+1), . . . , κ2(wm). The claim follows fromLemma2.1. �

Using Corollary10.3, we obtain the following corollary.

Corollary 10.6. Let G = 〈x1, . . . , xn | r1, . . . , rm〉 be a commutator-relators group, andh =h(G, k). Then, for each i≥ 2, the Lie algebrah/h(i) is isomorphic to the quotient of the free Liealgebralie(x) by the sum of the ideals(M2(r1), . . . ,M2(rm)) andlie(i)(x).

Now supposeG is 1-formal. Then, in view of Corollary6.6, the Chen Lie algebra gr(G/G′′; k)is isomorphic toh(G; k)/h(G; k)(i), which has presentation as above.


10.3. Koszul properties. We now use our presentation of the holonomy Lie algebrah = h(G, k)of a groupG with finitely many generators and relators to study the Koszul properties of thecorresponding universal enveloping algebra.

Computing the Hilbert Series ofU(h) directly from Corollary10.2 is not easy, since it in-volves finding a Grobner basis for a non-commutative algebra. However, ifU(h) is a Koszulalgebra, we can use Proposition3.7and the Corollary3.8 to reduce the computation to that ofthe Hilbert series of a graded-commutative algebra, which can be done by a standard Grobnerbasis algorithm.

Proposition 10.7. Let G be a group with presentation〈x | r 〉, and set m= |r |. If rank(ǫi(r j)) = mor m− 1, then the universal enveloping algebra U(h(G; k)) is Koszul.

Proof. Let h = h(G, k). By Proposition8.9and Theorem10.1,

(80) h =

lie(x) if rank(ǫi(r j)) = m,

lie(y)/ideal(κ2(wm)) if rank(ǫi(r j)) = m− 1.

In the first case,U(h) = T〈x〉, which of course is Koszul. In the second case, Corollary10.2implies thatU(h) = T〈y〉/I , whereI = ideal(κ2(wm)). Clearly,I is a principal ideal, generatedin degree 2; thus, by [23], U(h) is again a Koszul algebra. �

Of course, the universal enveloping algebra of the holonomyLie algebra of a finitely gener-ated group is a quadratic algebra. In general, though, it is not a Koszul algebra (see for instanceExample7.12.)

Example 10.8. Let h be the holonomy Lie algebra of the McCool groupPΣ+n . As shown byConner and Goetz in [14], the algebraU(h) is not Koszul forn ≥ 4. For more information onthe Lie algebras associated to the McCool groups, we refer to[67].

11. Mildness, one-relator groups, and link groups

We start this section with the notion of mild (or inert) presentation of a group, due to J. Labuteand D. Anick, and its relevance to the associated graded Lie algebra. We then continue withvarious applications to two important classes of finitely presented groups: one-relator groupsand fundamental groups of link complements.

11.1. Mild presentations. Let F be a finitely generated free group, with generating setx ={x1, . . . , xn}. Theweightof a wordr ∈ F is defined asω(r) = sup{k | r ∈ ΓkF}. SinceF isresidually nilpotent,ω(r) is finite. The image ofr in grω(r)(F) is called theinitial form of r, andis denoted by in(r).

Let G = F/R be a quotient ofF, with presentationG = 〈x | r 〉, wherer = {r1, . . . , rm}. Letink(r ) be the ideal of the freek-Lie algebralie(x) generated by{in(r1), . . . , in(rm)}. Clearly, thisis a homogeneous ideal; thus, the quotient

(81) Lk(G) := lie(x)/ ink(r )

is a graded Lie algebra. As noted by Labute in [31], the ideal ink(r ) is contained in grΓ(R; k),where ΓkR = ΓkF ∩ R is the induced filtration onR. Hence, there exists an epimorphismLk(G)։ gr(G; k).

Proposition 11.1.For every commutator-relators group G, the canonical projectionΦG : h(G; k)։gr(G, k) factors through an epimorphismh(G; k)։ Lk(G).


Proof. Let G = 〈x | r 〉 be a commutator-relators presentation for our group. By Corollary 10.3,the holonomy Lie algebrah(G; k) admits a presentation of the formlie(x)/a, wherea is the idealgenerated by the degree 2 part ofM(r) − 1, for all r ∈ r . On the other hand, in(r) is the smallestdegree homogeneous part ofM(r) − 1. Hence,a ⊆ ink(r ), and this complete the proof. �

Following [31, 1], we say that a groupG is amildly presented group(overk) if it admits a pre-sentationG = 〈x | r 〉 such that the quotient ink(r )/[ink(r ), ink(r )], viewed as aU(Lk(G))-modulevia the adjoint representation ofLk(G), is a free module on the images of in(r1), . . . , in(rm). Asshown by Anick in [1], a presentationG = 〈x1, . . . , xn | r1, . . . rm〉 is mild if and only if

(82) Hilb(U(Lk(G), t) =

1− nt+m∑

i=1

tω(r i )

−1

.

Theorem 11.2(Labute [30, 31]). Let G be a finitely-presented group.

(1) If G is mildly presented, thengr(G; k) = Lk(G).(2) If G has a single relator r, then G is mildly presented. Moreover, for each k≥ 1, the lcs

rankφk = dimk gr(G; k) is given by

(83) φk =1k

∑

d|k

µ(k/d)

∑

0≤i≤[d/e]

(−1)id

d+ i − ei

(d+ i − ie

i

)nd−ei

,

whereµ is the Mobius function and e= ω(r).

Labute states this theorem overZ, but his proof works for any commutative PID with unity.There is an example in [31] showing that the mildness condition is crucial for part (1) of thetheorem to hold. We give now a much simpler example to illustrate this phenomenon.

Example 11.3. Let G = 〈x1, x2, x3 | x3, x3[x1, x2]〉. Clearly,G � 〈x1, x2 | [x1, x2]〉, whichis a mild presentation. However, the Lie algebralie(x1, x2, x3)/ideal(x3) is not isomorphic togr(G; k) = lie(x1, x2)/ideal([x1, x2]). Hence, the first presentation is not a mild.

11.2. Mildness and graded formality. We now use Labute’s work on the associated gradedLie algebra and our presentation of the holonomy Lie algebrato give two graded-formalitycriteria.

Corollary 11.4. Let G be a group admitting a mild presentation〈x | r 〉. If ω(r) ≤ 2 for eachr ∈ r , then G is graded-formal.

Proof. By Theorem11.2, the associated graded Lie algebra gr(H, k) has a presentation of theform lie(x)/ ink(r ), with ink(r ) a homogeneous ideal generated in degrees 1 and 2. Using thedegree 1 relations to eliminate superfluous generators, we arrive at a presentation with onlyquadratic relations. The desired conclusion follows from Lemma4.10. �

An important sufficient condition for mildness of a presentation was given by Anick [1].Recall thatι denotes the canonical injection from the free Lie algebralie(x) into k〈x〉. Fix an or-dering on the set{x}. The set of monomials in the homogeneous elementsι(in(r1)), . . . , ι(in(rm))inherits the lexicographic order. Letwi be the highest term ofι(in(r i)) for 1 ≤ i ≤ m. Supposethat (i) nowi equals zero; (ii) nowi is a submonomial of anyw j for i , j, i.e.,w j = uwiv cannotoccur; and (iii) nowi overlaps with anyw j , i.e.,wi = uvandw j = vwcannot occur unlessv = 1,or u = w = 1. Then, the set{r1, . . . , rn} is mild (overk). We use this criterion to provide anexample of a finitely-presented groupG which is graded-formal, but not filtered-formal.


Example 11.5. Let G be the group with generatorsx1, . . . , x4 and relatorsr1 = [x2, x3], r2 =

[x1, x4], and r3 = [x1, x3][ x2, x4]. Ordering the generators asx1 ≻ x2 ≻ x3 ≻ x4, we findthat the highest terms for{ι(in(r1)), ι(in(r2)), ι(in(r3))} are {x2x3, x1x4, x1x3}, and these wordssatisfy the above conditions of Anick. Thus, by Theorem11.2, the Lie algebra gr(G; k) isthe quotient oflie(x1, . . . , x4) by the ideal generated by [x2, x3], [x1, x4], and [x1, x3] + [x2, x4].Hence,h(G; k) � gr(G; k), that is,G is graded-formal. On the other hand, using the TangentCone theorem from [17], one can show that the groupG is not 1-formal. Therefore,G is notfiltered-formal.

11.3. One-relator groups. If the groupG admits a finite presentation with a single relator,much more can be said.

Corollary 11.6. Let G= 〈x | r〉 be a1-relator group.

(1) If r is a commutator relator, thenh(G; k) = lie(x)/ideal(M2(r)).(2) If r is not a commutator relator, thenh(G; k) = lie(y1, . . . , yn−1).

Proof. Part (1) follows from Corollary10.3. Whenr is not a commutator relator, the Jacobianmatrix JG = (ǫ(∂ir)) has rank 1. Part (2) then follows from Theorem10.1. �

Corollary 11.7. Let G= 〈x1, . . . xn | r〉 be a1-relator group, and leth = h(G, k). Then

(84) Hilb(U(h); t) =

1/(1− (n− 1)t) if ω(r) = 1,

1/(1− nt+ t2) if ω(r) = 2,

1/(1− nt) if ω(r) ≥ 3.

Proof. Let x = {x1, . . . , xn}. By Corollary11.6, the universal enveloping algebraU(h) is iso-morphic to eitherT(y1, . . . , yn−1) if ω(r) = 1, or toT(x)/ ideal(M2(r)) if ω(r) = 2, or toT(x) ifω(r) ≥ 3. The claim now follows from Proposition3.7and Corollary3.8. �

Theorem 11.8.Let G= 〈x | r〉 be a group defined by a single relation. Then G is graded-formalif and only ifω(r) ≤ 2.

Proof. By Theorem11.2, the given presentation ofG is mild. The weightω(r) can also becomputed asω(r) = inf{|I | | M(r)I , 0}. If ω(r) ≤ 2, then, by Corollary11.6, we have that

(85) h(G; k) � gr(G; k) � lie(x)/ideal(in(r)),

and soG is graded-formal.On the other hand, ifω(r) ≥ 3, thenh(G; k) = lie(x). However, gr(G; k) = lie(x)/ideal(in(r)).

Hence,G is not graded-formal. �

Example 11.9. Let G = 〈x1, x2 | r〉, wherer = [x1, [x1, x2]]. Clearly,ω(r) = 3. Hence,G is notgraded-formal.

However, even if a one-relator group has weight 2 relation, the group need not be filtered-formal, as we show in the next example.

Example 11.10. Let G = 〈x1, . . . , x5 | [x1, x2][ x3, [x4, x5]] = 1〉. By Theorem11.2, the Liealgebra gr(G; k) has presentationlie(x1, . . . , x5)/ideal([x1, x2]). Thus, by Corollary11.4, thegroupG is graded-formal. On the other hand, Remark9.8 shows thatG admits a non-trivialtriple Massey product of the form〈u3, u4, u5〉. Thus,G is not 1-formal, and soG is not filtered-formal.


We now determine the ranks of the (rational) Chen Lie algebraassociated to an arbitraryfinitely presented, 1-relator, 1-formal group, thereby extending a result from [53].

Proposition 11.11.Let G= F/〈r〉 be a one-relator group, where F= 〈x1, . . . , xn〉, and supposeG is1-formal. Then

Hilb(gr(G/G′′, k), t) =

1+ nt−1− nt+ t2

(1− t)nif r ∈ [F, F],

1+ (n− 1)t −1− (n− 1)t(1− t)n−1

otherwise.

Proof. The first claim is proved in [53, Theorem 7.3]. To prove the second claim, recall thatgr(G/G′′, k) � h(G; k)/h(G; k)′′, for any 1-formal groupG. Now, since we are assuming thatr < [F, F], Theorem10.1implies thath(G; k) � lie(y1, . . . , yn−1). The claim follows from Chen’sformula (41) and the fact that gr(F/F′′, k) = lie(x1, . . . , xn)/lie′′(x1, . . . , xn), �

11.4. Link groups. We conclude this section with a very well-studied class of groups whichoccurs in low-dimensional topology. LetL = (L1, . . . , Ln) be ann-component link inS3. Thecomplement of the link,X = S3 \

⋃ni=1 Li , has the homotopy type of a connected, finite, 2-

dimensional CW-complex withb1(X) = n andb2(X) = n−1. The link group,G = π1(X), carriescrucial information about the homotopy type ofX: if n = 1 (i.e., the link is a knot), or ifn > 1andL is a not a split link, thenX is aK(G, 1).

Every link L as above arises as a closed-up braidβ. That is, there is a braidβ in the Artinbraid groupBk such thatL is isotopic to the link obtained fromβ by joining the top and bottomof each strand. The link group, then, has presentationG = 〈x1, . . . xk | β(xi) = xi (i = 1, . . . , k)〉,whereBk is now viewed as a subgroup of Aut(Fk) via the Artin embedding. Ifβ belongs to thepure braid groupPn ⊂ Bn, thenL = β is ann-component link, called apure braid link.

Associated to ann-component linkL there is a linking graphΓ, with vertex set{1, . . . , n},and an edge (i, j) for each pair of components (Li , L j) with non-zero linking number. Supposethe graphΓ is connected. Then, as conjectured by Murasugi [51] and proved by Massey–Traldi[47] and Labute [32], the link groupG has the same LCS ranksφk and the same Chen ranksθk

as the free groupFn−1, for all k > 1. Furthermore,G has the same Chen Lie algebra asFn−1 (see[53]). The next theorem is a combination of results from [3, 1, 53]

Theorem 11.12.Let L be an n-component link in S3 with connected linking graphΓ, and let Gbe the link group. Then

(1) The group G is graded-formal.(2) If L is a pure braid link, then G admits a mild presentation.(3) There exists a graded Lie algebra isomorphismgr(G/G′′; k) � h(G; k)/h(G; k)′′.

Proof. Part (1) follows from Lemma 4.1 and Theorems 3.2 and 4.2 in [3]. Part (2) is Theorem3.7 from [1], while Part (2) is proved in Theorem 10.1 from [53]. �

In general, though, a link group (even a pure braid link group) is not 1-formal. This phe-nomenon was first detected by W.S. Massey by means of his higher-order products [46], but thegraded and especially filtered formality can be even harder to detect.

Example 11.13.Let L be the Borromean rings. This is the 3-component link obtained by clos-ing up the pure braid [A1,2,A2,3] ∈ P′3, whereAi, j denote the standard generators of the pure braidgroup. All the linking numbers are 0, and so the graphΓ is disconnected. It is readily seen thatlink groupG passes Anick’s mildness test; thus gr(G; k) = lie(x, y, z)/ ideal([x, [y, z]] , [z, [y, x]]),


thereby recovering a computation of Hain [24]. It follows that G is not graded-formal, andthus not 1-formal. Alternatively, the non-1-formality ofG can be detected by the triple Masseyproducts〈u, v,w〉 and〈w, v, u〉.

Example 11.14.Let L be the Whitehead link. This is a 2-component link with linking number 0.Its link group is the 1-relator groupG = 〈x, y | r〉, wherer = x−1y−1xyx−1yxy−1xyx−1y−1xy−1x−1y.By Theorem11.2, this presentation is mild. Direct computation shows that in(r) = [x, [y, [x, y]]],and so gr(G; k) = lie(x, y)/ ideal([x, [y, [x, y]]]), again verifying a computation from [24]. Inparticular,G is not graded-formal. The non-1-formality ofG can also be detected by suitablefourth-order Massey products.

We do not know whether the two link groups from above are filtered-formal. Nevertheless,we give an example of a link group which is graded-formal, yetnot filtered-formal.

Example 11.15.Let L be the link of great circles inS3 corresponding to the arrangement oftransverse planes through the origin ofR4 denoted asA (31425) in [49]. ThenL is a pure braidlink of 5 components, with linking graph the complete graphK5; moreover, the link groupGis isomorphic to the semidirect productF4 ⋊α F1, whereα = A1,3A2,3A2,4 ∈ P4. By Theorem11.12, the groupG is graded-formal. On the other hand, as noted in [17, Example 8.2], theTangent Cone theorem does not hold for this group, and thusG is not 1-formal. Consequently,G is not filtered-formal.

12. Seifert fibered manifolds

We now use our techniques to study the fundamental groups of orientable Seifert manifoldsfrom a rational homotopy viewpoint. We start our analysis with the fundamental groups ofRiemann surfaces.

12.1. Riemann surfaces.LetΣg be the closed, orientable surface of genusg. The fundamentalgroupΠg = π1(Σg) is a 1-relator group, with generatorsx1, y1, . . . , xg, yg and a single relation,[x1, y1] · · · [xg, yg] = 1. The cohomology algebraA = H∗(Σg; k) is the quotient of the exterioralgebra on generatorsa1, b1, . . . , ag, bg, in degree 1 by the ideal generated byaibi − a jb j , for1 ≤ i < j ≤ g, together withaia j , bib j, aib j , a jbi , for 1 ≤ i < j ≤ g. Consequently,A is aquadratic algebra, and

(86) h(Πg; k) = lie(2g)/⟨ g∑

i=1

[xi , yi ] = 0⟩,

wherelie(2g) := lie(x1, y1, . . . , xg, yg).The Riemann surfaceΣg is a compact Kahler manifold, and thus it is a formal space. Hence,

the cohomology algebraA = H∗(Σg; k) with differentiald = 0 is a quadratic model forΣg.Furthermore,Πg is a 1-formal group, and so gr(Πg; k) = h(Πg; k).

It is readily seen that the defining ideal ofA admits a quadratic Grobner basis; therefore,Ais a Koszul algebra. Furthermore, Hilb(A, t) = 1 + 2gt + t2, and so Corollary4.5 implies that∏

k≥1(1 − tk)φk(Πg) = 1 − 2gt + t2. In fact, it follows from formula (83) that the lower centralseries ranks of the 1-relator groupΠg are given by

(87) φk(Πg) =1k

∑

d|k

µ(k/d)

[d/2]∑

i=0

(−1)id

d− i

(d− i

i

)(2g)d−2i

.


Using now Theorem10.5, we see that the Chen Lie algebra ofΠg has presentation

(88) gr(Πg/Π

′′g , k

)= lie(2g)

/(⟨ g∑

i=1

[xi , yi]⟩+ lie′′(2g)

).

Furthermore, Proposition11.11shows that the Chen ranks are given byθ1(Πg) = 2g, θ2(Πg) =2g2 − g− 1, and

(89) θk(Πg) = (k− 1)

(2g+ k− 2

k

)−

(2g+ k− 3

k− 2

), for k ≥ 3.

12.2. Seifert fibered spaces.We will consider here only orientable, closed Seifert manifoldswith orientable base. Every such manifoldM admits an effective circle action, with orbit spacean orientable surface of genusg, and finitely many exceptional orbits, encoded in pairs of co-prime integers (α1, β1), . . . , (αs, βs) with α j ≥ 2. The obstruction to trivializing the bundleη : M → Σg outside tubular neighborhoods of the exceptional orbits isgiven by an integerb = b(η). A standard presentation for the fundamental group ofM in terms of the Seifertinvariants is given by

πη := π1(M) =⟨x1, y1, . . . , xg, yg, z1, . . . , zs, h | h central,

[x1, y1] · · · [xg, yg]z1 · · · zs = hb, zαii hβi = 1 (i = 1, . . . , s)

⟩.

(90)

For instance, ifs= 0, the corresponding manifold,Mg,b, is theS1-bundle overΣg with Eulernumberb. Let πg,b := π1(Mg,b) be the fundamental group of this manifold. Ifb = 0, thenπg,0 = Πg × Z, whereas ifb = 1, then

(91) πg,1 = 〈x1, y1, . . . , xg, yg, h | [x1, y1] · · · [xg, yg] = h, h central〉.

In particular,M1,1 is the Heisenberg 3-dimensional nilmanifold andπ1,1 is the group from Ex-ample5.7.

12.3. Malcev Lie algebra. As shown in [63], the Euler numbere(η) of the Seifert bundleη : M → Σg satisfies

(92) e(η) = −b(η) −s∑

i=1

βi/αi .

The next result describes the minimal model of an (orientable) Seifert fibered space, as con-structed by G. Putinar in [60].

Theorem 12.1([60]). Let η : M → Σg be an orientable Seifert fibered space, with g> 0. Theminimal modelM (M) is (up to homotopy) the Hirsch extensionM (Σg) ⊗ (

∧(c), d), where the

differential is given by d(c) = 0 if e(η) = 0, and d(c) = generator of H2(Σg;Q) if e(η) , 0.

To each orientable Seifert bundleη : M → Σg, let us associate theS1-bundleη : Mg,ǫ(η) → Σg,whereǫ(η) = 0 if e(η) = 0, andǫ(η) = 1 if e(η) , 0.

Corollary 12.2. Letη : M → Σg be an orientable Seifert fibered space. The Malcev Lie algebraof the fundamental groupπη = π1(M) is given bym(πη; k) � m(πg,ǫ(η); k).

Proof. Wheng > 0, the above theorem implies thatM (M) = M (Mg,ǫ(η)), and the claim fol-lows. Wheng = 0, we have thatb1(πη) = 1 if ǫ(η) = 0 andb1(πη) = 0 if ǫ(η) = 1, and the claimagain follows. �


To avoid trivialities, we will assume thatg > 0 for the rest of this section. Using nowTheorem12.1, we obtain a quadratic model for the Seifert manifoldM, provided the baseΣg

has positive genus.

Lemma 12.3. Suppose g> 0. Then M has a quadratic model of the form(H∗(Σg; k)⊗

∧(c), d

),

wheredeg(c) = 1 and the differential d is defined by d(ai) = d(bi) = 0 for 1 ≤ i ≤ g, d(c) = 0 ife(η) = 0, and d(c) = a1 ∧ b1 if e(η) , 0.

Next, we will give an explicit presentation for the Malcev Lie algebram(πη; k) as the degreecompletion of a certain graded Lie algebra.

Theorem 12.4.The Malcev Lie algebra ofπη is the degree completion of the graded Lie algebra

(93) L(πη) =

lie(x1, y1, . . . , xg, yg, z)/〈

∑gi=1[xi , yi ] = 0, zcentral〉 if e(η) = 0;

lie(x1, y1, . . . , xg, yg,w)/〈∑g

i=1[xi , yi ] = w, w central〉 if e(η) , 0,

where,deg(w) = 2 and the other generators have degree1. Moreover,gr(πη; k) � L(πη).

Proof. If e(η) = 0, Corollary12.2says thatm(πη; k) is isomorphic to the Malcev Lie algebra ofπg,0 = Πg × Z, which is a 1-formal group. Furthermore, we know from (86) that gr(Πg; k) is thequotient of the free Lie algebralie(2g) by the ideal generated by

∑gi=1[xi , yi ]. Hence,m(πη; k) is

isomorphic to the degree completion of gr(Πg×Z) = gr(Πg; k)× gr(Z; k), which is precisely theLie algebraL(πη) from (93).

If e(η) , 0, Lemma12.3provides a quadratic model for our Seifert manifold. Takingthe Liealgebra dual to this quadratic model and using [6, Theorem 4.3.6], we obtain that the MalcevLie algebram(πη) is isomorphic to the degree completion of the graded Lie algebraL(πη).Furthermore, by the main theorem from [61], there is an isomorphism gr(m(πη; k)) � gr(πη; k).This completes the proof. �

Corollary 12.5. Fundamental groups of orientable Seifert manifolds are filtered-formal.

12.4. Holonomy Lie algebra. We now give a presentation for the holonomy Lie algebra of aSeifert manifold group.

Theorem 12.6.Letη : M → Σg be a Seifert fibration. The rational holonomy Lie algebra of thegroupπη = π1(M) is given by

h(πη; k) =

lie(x1, y1, . . . , xg, yg, h)/〈

∑si=1[xi , yi ] = 0, h central〉 if e(η) = 0;

lie(2g) if e(η) , 0.

Proof. First assumee(η) = 0. In this case, the row-echelon approximation ofπη has presentation

πη = 〈x1, y1, . . . , xg, yg, z1, . . . , zs, h | zαii hβi = 1 (i = 1, . . . , s),

([x1, y1] · · · [xg, yg])α1···αs = 1, h central〉

(94)

It is readily seen that the rank of the Jacobian matrix associated to this presentation has ranks.Furthermore, the mapπ : Fk → Hk is given byxi 7→ xi , yi 7→ yi , zj 7→ (−βi/αi)h, h 7→ h. Let κbe the Magnus expansion from Definition8.2. A Fox Calculus computation shows thatκ takesthe following values on the commutator-relators ofπη:

κ(r) = 1+ (α1 · · ·αs)(x1y1 − y1x1 + · · · + xgyg − ygxg) + terms of degree≥ 3,

κ([xi, h]) = 1+ xih− hxi + terms of degree≥ 3,

κ([yi, h]) = 1+ yih− hyi + terms of degree≥ 3,

κ([yi, z]) = 1+ terms of degree≥ 3,


wherer = ([x1, y1] · · · [xg, yg])α1···αs. The first claim now follows from Theorem10.1.Next, assumee(η) , 0. Then the row-echelon approximation ofπη is given by

πη = 〈x1, y1, . . . , xg, yg, z1, . . . , zs, h | zαii hβi = 1 (i = 1, . . . , s),

([x1, y1] · · · [xg, yg])α1···αshe(η)α1···αs = 1, h central〉,(95)

while the homomorphismπ : Fk → Hk is given byxi 7→ xi , yi 7→ yi , zj 7→ (−βi/αi)h, h 7→ 0. Asbefore, the second claim follows from Theorem10.1, and we are done. �

12.5. LCS ranks. We end this section with a computation of the ranks of the various gradedLie algebras attached to the fundamental group of a Seifert manifold. Comparing these ranks,we derive some consequences regarding the non-formality properties of such groups.

We start with the LCS ranksφk(πη) = dim grk(πη; k) and the holonomy ranks are defined asφk(πη) = dim(h(πη; k)k.

Proposition 12.7. The LCS ranks and the holonomy ranks of a Seifert manifold group πη arecomputed as follows.

(1) If e(η) = 0, thenφ1(πη) = φ1(πη) = 2g+ 1, andφk(πη) = φk(πη) = φk(Πg) for k ≥ 2.(2) If e(η) , 0, thenφk(πη) = φk(F2g) for k ≥ 1.(3) If e(η) , 0, thenφ1(πη) = 2g,φ2(πη) = g(2g− 1), andφk(πη) = φk(Π2g) for k ≥ 3.

Here the LCS ranksφk(Πg) are given by formula(87).

Proof. If e(η) = 0, thenπη � Πg × Z, and claim (1) readily follows. So suppose thate(η) , 0.In this case, we know from Theorem12.6thath(πη; k) = h(F2g; k), and thus claim (2) follows.

By Theorem12.4, the associated graded Lie algebra gr(πη; k) is isomorphic to the quotient ofthe free Lie algebralie(x1, y1, . . . , xg, yg,w) by the ideal generated by the elements

∑gi=1[xi , yi ] −

w, [w, xi ], and [w, yi]. Define a morphismχ : gr(πη; k)→ gr(Πg; k) by sendingxi 7→ xi , yi 7→ yi ,andw 7→ 0. It is readily seen that the kernel ofχ is the Lie ideal of gr(πη; k) generated byw,and this ideal is isomorphic to the free Lie algebra onw. Thus, we get a short exact sequence ofgraded Lie algebras,

(96) 0 // lie(w) // gr(πη; k)χ // gr(Πg; k) // 0 .

Comparing Hilbert series in this sequence establishes claim (3) and completes the proof. �

Corollary 12.8. If g = 0, the groupπη is always1-formal, while if g > 0, the groupπη isgraded-formal if and only if e(η) = 0.

Proof. First supposee(η) = 0. In this case, we know from Theorem12.4 that gr(πη; k) �gr(Πg; k)× gr(Z; k). It easily follows that gr(πη; k) � h(πη; k) by comparing the presentations ofthese two Lie algebras. Hence,πη is graded-formal, and thus 1-formal, by Corollary12.5.

It is enough to assume thatg > 0 ande(η) , 0, since the other claims are clear. By Proposi-tion 12.7, we have thatφ3(πη) = (8g3 − 2g)/3, whereasφ3(πη) = (8g3 − 8g)/3. Hence,h(πη; k)is not isomorphic to gr(πη; k), proving thatπη is not graded-formal. �

12.6. Chen ranks. Recall that the Chen ranks are defined asθk(πη) = dim grk(πη/π′′η ; k), while

the holonomy Chen ranks are defined asθk(πη) = dim(h/h′′)k, whereh = h(πη; k).

Proposition 12.9. The Chen ranks and the holonomy Chen ranks of a Seifert manifold groupπη are computed as follows.

(1) If e(η) = 0, thenθ1(πη) = θ1(πη) = 2g+ 1, andθk(πη) = θk(πη) = θk(Πg) for k ≥ 2.


(2) If e(η) , 0, thenθk(πη) = θk(F2g) for k ≥ 1.(3) If e(η) , 0, thenθ1(πη) = 2g, θ2(πη) = g(2g− 1), andθk(πη) = θk(Π2g) for k ≥ 3.

Here the Chen ranksθk(F2g) andθk(Πg) are given by formulas(41) and (89), respectively.

Proof. Claims (1) and (2) are easily proved, as in Proposition12.7. To prove claim (3), start byrecalling from Corollary12.5that the groupπη is filtered-formal. Hence, by Theorem6.4, theChen Lie algebra gr(πη/π′′η ; k) is isomorphic to gr(πη; k)/ gr(πη; k)′′. As before, we get a shortexact sequence of graded Lie algebras,

(97) 0 // lie(w) // gr(πη/π′′η ; k) // gr(Πg/Π′′g ; k) // 0 .

Comparing Hilbert series in this sequence completes the proof. �

Remark 12.10. The above result can be used to give another proof of Corollary 12.8. Indeed,supposee(η) , 0. Then, by Proposition12.9, we have thatθ3(πη) − θ3(πη) = 2g. Consequently,by Corollary6.6, the groupπη is not 1-formal. Hence, by Corollary12.5, πη is not graded-formal.

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Department of Mathematics, Northeastern University, Boston, MA 02115, USAE-mail address: [email protected]: http://www.northeastern.edu/suciu/

Department of Mathematics, Northeastern University, Boston, MA 02115, USAE-mail address: [email protected]: http://myfiles.neu.edu/wang.he1/






















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Formality properties of finitely generated groups and Lie algebras

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