Various types of algebraic rings

8/11/2019 Various types of algebraic rings

http://slidepdf.com/reader/full/various-types-of-algebraic-rings 1/29

a r X i v : m a t h / 0 4 1 0 2 2 6 v 1 [ m a t h . R

A ] 8 O c t 2 0 0 4

BRANCH RINGS, THINNED RINGS, TREE ENVELOPINGRINGS

BILBO BAGGINS AND LAURENT BARTHOLDI

Abstract. We develop the theory of “branch algebras”, which are innite-dimensional associative algebras that are isomorphic, up to taking subrings of nite codimension, to a matrix ring over themselves. The main examples comefrom groups acting on trees.

In particular, we construct an algebra over the eld of two elements, thatis nitely generated, prime, innite-dimensional but with all proper quotientsnite, has a recursive presentation, is graded, and has Gelfand-Kirillov dimen-sion 2.

1. Introduction

Rings are powerful tools, and those arising from groups have been studied ingreat length [32, 39, 40]. The rst author’s long-awaited monograph on the topicshould prove illuminating [5].

Although rings arising from groups are very interesting from a ring theorists’perspective, they are in a sense “too large”, because some proper quotient of themmay still contain a copy of the original group. The process of “quotienting outextra material” from a group ring while retaining the original group intact is the“thinning process” described in [38].

In this paper, we consider a natural ring arising from a group acting on a rootedtree, which we call its “tree enveloping ring”. This is a re-expression, in termsof matrices, of Said Sidki’s construction [38]. If the group’s action has some self-similarity modeled on the tree’s self-similarity, we may expect the same to happenfor the associated ring, and we use this self-similarity as a leitmotiv for all ourresults.

Loosely speaking (See §3.1.6 for a more precise statement), a weakly branchalgebra is an algebra A such that (1) there is an embedding ψ : A → M q(A) forsome q , and (2) for any n there is an element of A such that ψn (a) has a singlenon-zero entry. We show (Theorem 3.10) that such algebras may not satisfy apolynomial identity.

The main construction of weakly branch algebras is via groups acting on trees;the algebra A is then the linear envelope of the groups’ linear representation on the

boundary of the tree. We show (Theorem 3.22) that if the groups’ orbits on theDate : Last revised October 7, 2004; typeset May 6, 2013.2000 Mathematics Subject Classication. 20E08 (Groups acting on trees), 16S34 (Group

rings), 17B50 (Modular Lie algebras), 11K55 (Hausdorff dimension).Key words and phrases. Groups acting on trees. Branch groups. Graded rings. Gelfand-

Kirillov dimension.The rst author acknowledges limited support from the Brandywine county.The second author acknowledges support from TU Graz and UC Berkeley, where part of this

research was conducted.1

http://arxiv.org/abs/math/0410226v1








































2 BILBO BAGGINS AND LAURENT BARTHOLDI

boundary have polynomial growth of degree d, then the Gelfand-Kirillov dimensionof A is at most 2d. In particular contracting groups generate algebras of niteGelfand-Kirillov dimension.

We next concentrate in more detail on the rings A arising from the Grigorchukgroup [18] G. Recall that G is a just-innite, nitely generated torsion group. Thealgebra A over the eld F2 was already studied by Ana Cristina Vieira in [41]. Thefollowing theorem summarizes our results:

Theorem 1.1. The ring A is just-innite and prime (Theorem 4.3). It is recur-sively presented (Theorems 4.5 and 4.14), and has Gelfand-Kirillov dimension 2(Theorem 4.6 and Corollary 4.17). The ring A has an ideal J , and an embed-ding ψ : A → M 2(A), such that all the following: ψ−1 : M 2( J ) → J , J → A,ψ : A →M 2(A) are inclusions with nite cokernel 1 (Theorem 4.3).

Over a eld of characteristic 2, the ring A is graded (Corollary 4.15, and may be presented as

A = A,B,C,D |A2 , B 2 , C 2 , B + C + D,BC,CB,DAD,σn (CACACAC ), σn (DACACAD ) for all n ≥0 ,

where σ is the substitution σ : S →S dened by

A →ACA, B →D, C →B, D →C.

The subgroup generated by {1+ A, 1+ B, 1+ C, 1+ D}is isomorphic to the Grigorchuk group G. The ring A also contains a copy of the Laurent polynomials F2[X, X −1](Theorem 4.19).

1.1. Plan. Section 2 recalls constructions and results concerning groups acting onrooted trees. A few of the results are new (Propositions 2.6 and 2.8); the othersare given with brief proofs, mainly to illustrate the parallelism between groups andalgebras.

Section 3 introduces branch algebras, and develops general tools and resultsconcerning them; in particular, the branch algebra associated with a group actingon a rooted tree.

Section 4 studies more intricately the branch algebra associated with the Grig-orchuk group. Its study then splits in two cases, depending on the characteristicbeing tame ( = 2) or wild (= 2). More results hold in characteristic 2, in particularthe branch algebra is graded; some results hold in both cases but the proofs aresimpler in characteristic 2, and therefore are given in greater detail there.

1.2. Notation. We use the following notational conventions: functions are writtenx →xf if they are part of a group that acts, and x →f (x) otherwise. All groups

are written in usual capitals ( G), and algebras in gothic ( A). We use ε for theaugmentation map on group rings, = ker ε for the augmentation ideal, and R forthe Jacobson radical.

1.3. Thanks. I am greatly indebted to Katia Pervova, Said Sidki and Em Zel-manov for their generous discussions on this topic.

1i.e. the image has nite codimension in the target



BRANCH RINGS, THINNED RINGS, TREE ENVELOPING RINGS 3

2. Groups acting on trees

2.1. Groups and trees. We start by reviewing the basic notions associated to

groups acting on rooted trees.

2.1.1. Trees. Let X be a set of cardinality # X ≥2, called the alphabet . The regular rooted tree on X is X , the set of (nite) words over X . It admits a natural treestructure by putting an edge between words of the form x1 . . . xn and x1 . . . x n xn +1 ,for arbitrary xi X . The root is then the empty word.

More pedantically, the tree X is the Hasse diagram of the free monoid X onX , ordered by right divisibility ( v ≤w u : vu = w).

Let G be a group with given action on a set X . Recall that A G, the wreath product of A with G, denotes the group AX G, or again pedantically the semi-direct product with G of the sections of the trivial A-bundle over X .

2.1.2. Decomposition. Let W = Aut X be the group of graph automorphisms of

X . For each n N, the subset X n

of X is stable under W , and is called the n th layer of the tree. The group W admits a natural map, called the decomposition

φ : W →W S X ,

given by φ(g) = ( f, π g ) where πg S X , the activity of g, is the restriction of g tothe subset X X , and f : X → W is dened by xπ g wf (x ) = ( xw)g , or in otherwords f (x) is the compositum X → xX g

→ πg (x)X → X , where the rst andlast arrows are given respectively by insertion and deletion of the rst letter.

The decomposition map can be applied, in turn, to each of the factors of W S X .By abuse of notation, we say that we iterate the map φ on W , yielding φ2 : W →W S X S X ≤ W S X 2 , etc. More generally, we write φn : W → W S X n , andπn its projection to S X n .

The action of W on X uniquely extends, by continuity, to an action on X ω , the(Cantor) set of innite sequences over X . The self-similarity of X ω is expressed viathe decomposition X ω = x X X ω . This gives, for all n N, a continuous mapX ω →X n obtained by truncating a word to its rst n letters.

2.1.3. X -bimodule. There is a left-action and a right-action @ of the free monoidX on W , dened for x X and g W by

x g : w →x(vg ) if w = xvw otherwise ,

and g@x : w →v if (xw)g = xg v.

These actions satisfy the following properties:

(g@v)@w = g@(vw), v (w g) = ( vw) g,(1)(gh)@v = ( g@v)(h@vg ), v (gh) = ( v g)(v h),(2)

g = ( v g)@v, g = v X n

v (g@v) πng ,(3)

where in the last expression the v (g@v) mutually commute when v ranges overthe n th layer X n .

In this terminology, when we wrote the decomposition as φ(g) = ( f, π g ), we hadf (x) = g@x.




2.1.4. Branchness. Let G < W be a group acting on the regular rooted tree X .The vertex stabilizer Stab G (v) is the subgroup of G xing v X . The group G is

level-transitive , if G acts transitively on X n

for all n N;weakly recurrent , if it is level-transitive, and Stab G (x)@x < G for all x X ;recurrent , if it is level-transitive, and Stab G (x)@x = G for all x X ;weakly branch , if G is level-transitive, and ( v G) ∩G is non-trivial for all

v X ;weakly regular branch , if G is level-transitive, and has a non-trivial normal

subgroup K , called the branching subgroup , with x K < K for all x X ;branch , if G is level-transitive, and (v G) ∩G : v X n has nite index in

G for all n N;regular branch , if G is level-transitive, and has a nite-index normal sub-

group K with x K < K for all x X ;Weak branchness can be reformulated in terms of the action on X ω . Then G

is weakly branch if every closed set F X ω has a non-trivial xator Fix G (F ) =

{g G |g(f ) = f f F }.Remark that if G is branch, then K X has nite index in φ(K ), because it has

nite index in GX and in G S X .Remark also that if G is weakly regular branch, then there is a unique maximal

branching subgroup K ; it is

K =v X

(G ∩(v G))@v.

Proposition 2.1. If G is transitive, then it acts transitively on each layer of X ;therefore its action on (X ω , Bernoulli ) is ergodic. In particular, G is innite.Proof. Proceed by induction on n. Consider a layer X n of the tree, and two verticesx1 . . . x n and y1 . . . yn . Since G is branch, it acts transitively on X , so x1 . . . xn andy1x2 . . . x n belong to the same orbit. By induction, x2 . . . xn and y2 . . . yn are in thesame G-orbit; therefore, since G is recurrent, y1x2 . . . xn and y1 . . . yn belong to thesame orbit.

If the action is not ergodic, let A X ω be an invariant subset of non- {0, 1}measure. Then there exists n N such that X ω → X n is not onto; its image is aG-orbit, and thus the action of G is not transitive on the nth layer.

Proposition 2.2. If G is regular branch, then it is regular weakly branch and branch; if it it branch, then it is weakly branch; if it is regular weakly branch, then it is weakly branch.Proof. Let G be a regular branch group, with branching subgroup K . By Propo-sition 2.1, G is innite so K is non-trivial. This shows that G is regular weaklybranch. Assume now only that K is non-trivial, and let v X n be any vertex.Since K X

n

≤ φn (G), we may take any k

= 1 in K and consider the element

k v G. This shows that G is weakly branch. The other implications are of thesame nature.

Note nally that the group G is determined by a generating set S and therestriction of the decomposition map φ to S , in the following sense:Proposition 2.3. Let F be a group generated by a set S , and let φ : F →F S X be any map. Then there exists a unique subgroup G of W = Aut X that is generated by S and has decomposition map induced by φ through the canonical map F →G.




Proof. The decomposition map φ yields, by iteration, a map F → S X n for alln N. This denes an action on the n th layer of the tree X , and since they arecompatible with each other they dene an action of F on the tree. We let G bethe quotient of F by the kernel of this action. On the other hand, the action of thegenerators, and therefore of G, is determined by φ, so G is unique.

In particular, F may be the free group on S , and φ may be simply dened bythe choice, for each generator in S , of # X words and a permutation.

Therefore, in dening a recurrent group, we will only give a list of generators,and their images under φ. If X = {1, . . . , q }, we describe φ on generators with thenotation

φ(g) = g@1, . . . , g @q πg , or even φ(g) = g1 , . . . , gq if πg = 1 ,

rather than in the form φ(g) = ( f, π ) with f (x) = [email protected] that there may exist other groups G′ generated by S , and such that the

natural map F

→G′ induces an injective map G′

→G′ S X . However, such G′ will

not act faithfully on X . The group G dened by Proposition 2.3 is the smallest quotient of F through which the decomposition map factors.

Weakly branch groups are known to satisfy no group law. This follows from thefollowing general result, due to Mikl´ os Abert:

Proposition 2.4 ([2, Theorem 1]) . Let G be a group acting on a set X , such that for every nite Y X the xator 2 of Y does not x any point in X \Y . Then Gdoes not satsify any identity, i.e. for every w = 1 in the free group F (y1 , . . . , y k )there exist g1 , . . . , gk G with w(g1 , . . . , gk ) = 1 .

His proof goes as follows: let wi be the length- i prex of w, and let x X be any.Then, inductively on i, one shows that there exist g = ( g1 , . . . , gk ) Gk such thatx, x w1 (g) , . . . , x w i (g) are all distinct. The following is a weakening of Corollary 4in [2].

Corollary 2.5. If G is weakly branch, then it does not satisfy any group law.

Proof. Let G act on the boundary X ω of the tree X . Let Y X be a nite subset,and let ξ X \ Y be any. Then there exists a vertex v X on the geodesic ξ buton none of the geodesics in Y . Set K = G ∩(G v). Since G is weakly branch, K is non-trivial. Assume by contradiction that K xes ξ . Then since K is invariantunder the stabilizer of v, and G acts level-transitively, it follows that K also xesall images of ξ under the stabilizer of ξ ; this is a dense subset of vX ω , so K xesX ω , which contradicts the non-triviality of K . Therefore there exists g K withg|Y = 1 and ξ g = ξ , so the conditions of Proposition 2.4 are satised.

2.2. Dimension. Every countable residually- p group has a representation as asubgroup of Aut X , for X =

{1, . . . , p

}: x a descending ltration G = G0

≥G1

≥G2 ≥. . . with Gn = {1} and [Gn : Gn +1 ] = p; identify X with Gn /G n +1 . ThenG/G n is identied with X n , and G acts faithfully, by multiplication on cosets, onthe tree X . In general, this action will not be recurrent. Moreover, this action maybe “inefficient” in that the quotient of G represented by the action on X n may bequite small — if Gn G this quotient is G/G n of order pn , while the largest p-groupacting on X n has order p( pn −1) / ( p−1) . This motivates the following denition.

2aka “pointwise stabilizer”




Let W n = πn (W ) be quotient of W acting on X n . We give W the structure of a compact, totally disconnected metric space by setting

d(g, h) = inf {1/ # W n |πn (g) = πn (h)}.We obtain in this way the notion of closure and Hausdorff dimension . Explicitly,for a subgroup G < W , we have by [1]

Hdim( G) = liminf n →∞

log# πn (G)log# W n

;

see also [6]. The Hausdorff dimension of G coincides with that of its closure.

2.2.1. The tree closure. Let P ≤ S X be any group acting on X . The tree closure of P is the subgroup P of W consisting of all g W such that πn (g) n P ≤S X n

for all n N. It is the inverse limit of the groups n P , and is a closed subgroup of W .

We have P = P P , and πn (P ) = πn −1(P ) P , so # πn (P ) = (# πn −1 (P ))# X # P ,and therefore

(4) # πn (P ) = (# P )# X n − 1

# X − 1 .

In particular, # W n = (# S X )(# X n −1) / (# X −1) , and P has Hausdorff dimensionlog# P/ log(# X !).

If p is prime and X = {1, . . . , p}, we will often consider subgroups G of W p = P ,where P = (1, 2, . . . , p ) is a p-Sylow of S X . The dimension of G will be thencomputed relative to W p , by the simple formula

Hdim p(G) = HdimGHdim W p

= Hdim Glog( p!)log p

.

Proposition 2.6. Let G be a regular branch group. Then G has positive Hausdorff

dimension.If furthermore G is a subgroup of W p, then its relative Hausdorff dimension Hdim p is rational.

Proof. Let G have branching subgroup K , and for all n N set Gn = πn (G). LetM Nbe large enough so that G/φ −2(K X

2) maps isomorphically into GM /π M −2 (K X

2).

We then have, for all n ≥M ,

# Gn = [G : K ]# πn (K ) = [G : K ][φ(K ) : K X ](# πn −1 (K ))# X(5)

= [ G : K ]1−# X [φ(K ) : K X ](# Gn −1)# X .

Write log# Gn = α# X n + β , for some α, β to be determined; we have, again forn ≥M ,

α# X n + β = (1

−# X )log[G : K ] + log[φ(K ) : K X ] + # X (α# X n −1 + β ),

so β = log[G : K ]−log[φ(K ) : K X ]/ (# X −1). Then set α = (log # GM −β )/ # X M .We have solved the recurrence for # Gn , and α > 0 because Gn has unboundedorder.

Now it suffices to note that Hdim( G) = α(# X −1)/ log(# X !) to obtain Hdim( G) >0.

For the last claim, note that all indices in (5) are powers of p, and hence theirlogarithms in base p are integers.




Question 1. Mikl´ os Abert and B alint Vir´ ag [3] show that there exist free subgroups of W of Hausdorff dimension 1. Is there a nitely generated recurrent group of dimension 1? A branch group?

2.3. Growth. Let G be a group generated by a nite set S . The length of g Gis dened as g = min {n |g = s1 . . . s n for some s i S }. The word growth of Gis the function

f G,S (n) = # {g G | g ≤ n}.This function depends on the choice of generating set S . Given f , g : N → R, sayf g if there exists M N with f (n) ≤g(Mn ), and say f g if f g f ; thenthe equivalence class of f G,S is independent of S . The group G has exponential growth if f G,S en , and polynomial growth if f G,S n D for some D N. In allother cases, f G,S grows faster than any polynomial and slower than any exponential,and G has intermediate growth . If furthermore f G,S (n) ≥ An for some A > 1,uniformly on S , then G has uniformly exponential growth .

More generally, let E be a space on which G acts, and let E be any. Thenthe growth of E is the function

f E, ,S (n) = # {e E |e = g with g ≤ n}.

If E = G with left regular action, we recover the previous denition of growth. Wewill be interested in the case E = X ω with the natural action of G, or equivalentlyof E = G/ Stab G ( ) for some X ω .

2.3.1. Contraction. Let G be a nitely generated recurrent group. It is contracting if there exist λ < 1, n N and K such that, for all g G and v X n we have

g@v ≤ λ g + K .

Proposition 2.7 ([11], Proposition 8.11) . If G is contracting, then the growth of (X ω , ) is polynomial, of degree at most −n log# X/ log λ .

Conversely, if (X ω

, ) has polynomial growth of degree d, then G is contracting for any n large enough and any λ > (# X )−n/d .

Proposition 2.8. If G is a nitely generated branch group, and (X ω , ) has poly-nomial growth of degree d, then G has growth

f G (n) exp nd/ (d+1) .

Proof. Let us write q = # X . Let K be a branching subgroup, and set R0 =min { g | g K, g = 1 }. Let n N and v X n be given. For g K satisfying

g ≤ R0 , set hv,g = v g. By Proposition 2.7, we have hv,g ≤ q n/d g ≤ q n/d R0 .We now choose for all v X n some gv K with gv ≤ R0 , and consider the

corresponding elementh =

v X n

hv,g v .

On the one hand, there are at least 2 qnsuch elements, because there are at least 2

choices for each gv . On the other hand, such an element has length at most q n q n/d .If f (R) denote the growth function of G, we therefore have f (q n + n/d ) ≥2qn

, or inother words

f (R) exp q log R/ (1+ 1d )log q = exp Rd/ (d+1) .






The nite quotient πn (G ) has order 2 5·2n − 3 +2 , for n ≥ 3. It follows that G hasHausdorff dimension 5 / 8.

Igor Lysenok obtained in [28] a presentation of G by generators and relations:

Proposition 2.9 ([28]). Consider the endomorphism σ of {a,b,c,d} dened by

(6) a →aca, b →d, d →c, c →b.

Then

(7) G = a,b,c,d a2 , b2 , c2 , d2 ,bcd,σn (ad)4 , σn (adacac )4 n ≥0 .

2.4.3. The Gupta-Sidki group. This group Γ acts on the ternary tree, with X =

{1, 2, 3}. It is best described as the group generated by {x, γ }, with decompositions

φ(x) = 1, 1, 1 (1, 2, 3), φ(γ ) = γ,x,x −1 .

This group was studied by Narain Gupta and Said Sidki [23], who showed that Γis an innite 3-torsion group.

This group is contracting with n = 1 and λ = 12 .

The nite quotient π n (Γ) has order 3 2·3n − 1 +1 , for n ≥ 2. It follows that Γ hasHausdorff dimension 4 / 9 in W 3 .

The group Γ is branch, with branching subgroup Γ′ = [Γ, Γ]. Indeed φ(Γ′)contains Γ′ ×Γ′ ×Γ′, because φ([γ −1γ −x 2

, γ x γ ]) = 1, 1, [x, γ ] .Later Said Sidki constructed a presentation of Γ by generators and relations [37],

and associated an algebra to Γ — see Theorem 4.1.

2.4.4. Weakly branch groups. Most known examples of recurrent groups are weaklybranch. Among those that are not branch, one of the rst to be considered acts onthe ternary tree {1, 2, 3} :

Γ = x, δ given by φ(x) = 1, 1, 1 (1, 2, 3), φ(δ ) = δ,x,x ;

It was studied along with G , Γ and two other examples in [8, 9]. The nite quotientπn (Γ) has order 3

14 (3 n +2 n +3) , for n ≥2. It follows that Γ has Hausdorff dimension

1/ 2 in W 3 .Two interesting examples, acting on the binary tree, were also found:

2.4.5. The “BSV” group.

G1 = τ, µ given by φ(τ ) = 1, τ (1, 2), φ(µ) = 1, µ−1 (1, 2);

it was studied in [14], who showed that it is torsion-free, weakly branch, and con-structed a presentation of G1 . The nite quotient π2n (G1) has order 2

13 (2 2 n −1)+ n ,

for n ≥1. It follows that G1 has Hausdorff dimension 1 / 3.

2.4.6. The Basilica group.G2 = a, b given by φ(a) = 1, b (1, 2), φ(b) = 1, a ;

it was studied in [21], who showed that it is torsion-free and weakly branch, andin [13], who showed that it is amenable, though not “subexponentially elementaryamenable”. The nite quotient π2n (G2 ) has order 2

23 (2 2 n −1)+ n , for n ≥1. It follows

that G2 has Hausdorff dimension 2 / 3.All of these groups are contracting with n = 1 and λ = 1√ 2 .




2.4.7. The odometer. This is a group acting on {1, 2} :

Z = τ , φ(τ ) = 1, τ (1, 2).

Its action on the n th layer is via a 2 n -cycle. It is not weakly branch.2.4.8. The Lamplighter group. This is the group G = ( Z/ 2)( Z ) Z , the semidirectproduct with Z of nitely-supported Z/ 2-valued functions on Z. It acts on {1, 2} :

G = a, b , φ(a) = a, b (1, 2), φ(b) = a, b .

Again this group is not weakly branch.

2.4.9. More branch groups. The following general construction yields weakly branchgroups:Proposition 2.10. Let G be a contracting group. Then either G is virtually nilpo-tent, or G is weakly branch.Proof. A weakly branch group contains subgroups with arbitrarily large rank: pick

an arbitrary cut set F X , and for each f F pick a non-trivial element gf thatxes X \ f X . Then gf is an abelian subgroup of rank # F . This shows that aweakly branch group cannot be virtually nilpotent.

On the other hand, let G be a non-virtually nilpotent group. Then its wordgrowth is super-polynomial [22]. Since the Schreier graphs of G have polynomialgrowth [9], the projections GX n

GX n

∩φn (G) →GX n \{v} have non-trivial kernel,for every v X n ; so G is weakly branch.

3. Algebras

We consider various denitions of “recurrence” and “branchness” in the contextof algebras. Let k be a eld, xed throughout this section.

3.1. Associative algebras. If X is a set, we write M X (k) = M X the matrix

algebra of endomorphisms of the vector space kX , and for a k-algebra A we writeM X (A) = M X (k) A.

3.1.1. Recurrent transitive algebras. A recurrent transitive algebra is an associativealgebra A, given with an injective homomorphism ψ : A → M X (A), for some setX , such that for every x, y X the linear map A → M X (A) → A, obtained byprojecting ψ(A) on its (x, y ) matrix entry, is onto.

The map ψ is called the decomposition of A , and can be iterated, yielding a mapψn : A →M X n (A).

The most naive examples are as follows: consider the vector space V = kX ω ,and A = End( V ). The decomposition map is given by ψ : a →(ax,y ) where ax,y isdened on the basis vectors w X as follows: if a(xw) = bv v, then

ax,y (w) =v= yv ′ X ω

bv v′.

Similarly, consider the vector space V = kX ωof functions on X ω , and A =

End( V ). The decomposition map is given by

ψ(a) = ( ax,y ) where ax,y (f )(w) = a(v →f (xv))( yw).

These examples are meant to illustrate the connection between action on X ω andrecurrent algebras; they will not be considered below. However, all our algebraswill be subalgebras of these, i.e. contained in ωM X = M X ω .




3.1.2. Decomposition. Similarly to Proposition 2.3, a recurrent transitive algebramay be dened by its decomposition map, in the following sense:

Lemma 3.1. If F is an algebra generated by a set S , and ψ : F →M X (F) is a mapsuch that ψ(F)x,y = F for all x, y X , then there exists a unique minimal quotient of F that is a recurrent transitive algebra.

Proof. Set I 0 = ker ψ and I n +1 = ψ−1M X (I n ) for n N and I = n N I n . ThenI is an ideal in F, and F/ I is a recurrent transitive algebra. Consider the ideal Jgenerated by all ideals K ≤F such that ψ(K ) ≤M X (K ); then I ≤ J , and A = F / Jis the required minimal quotient of F .

It follows that a branch algebra may be dened by a choice, for each generatorin S , of # X 2 elements of the free algebra k S . Note that we do not mention anytopology on A ; if A is to be, say, in the category of C -algebras, then the denition

becomes much more intricate due to the absence of free objects in that category.The best approach is probably that of a C -bimodule considered in [29].An important feature is missing from the algebras of §3.1.1, namely the exis-

tence of nite-dimensional quotients similar to group actions on layers. These areintroduced as follows:

3.1.3. Augmented algebras. Let A be a recurrent transitive algebra. It is augmented if there exists a homomorphism ε : A →k, called the augmentation , and a subalge-bra P of M X with a homomorphism ζ : P →k, such that the diagram

Aψ

ε

ψ(A) ≤M X A

1 ε

k Pζ

commutes. We abbreviate “augmented recurrent transitive algebra” to art algebra ,or P -art algebra if we wish emphasize the P ≤M X used.

Let P be a subalgebra of M X , with augmentation ζ : P → k. There are twofundamental examples of art algebras, constructed as follows:

3.1.4. The “tree closure” P . We dene for all n N an augmented algebra P n ≤M X n , with ζ n : P n →k, for n N by P 1 = P , ζ 1 = ζ , and

P n +1 = m p M X P n |ζ n ( p)m P .

Its augmentation is given by ζ n +1 (m p) = ζ (ζ n ( p)m).Then there is a natural map P n +1 → P n , dened by m1 ·· · mn +1 →ζ (mn +1 )m1 · · · mn . We set P = lim

←−P n .

Then P is an art algebra: for a P , write a = lim←−an with an P n . Thenan +1 = mn pn with mn M X and pn P n . The sequence mn is constantequal to m, and we set ψ(a) = m lim←− pn .




The following diagram gives a natural map A →P for any P -art algebra A . Wewill always suppose that this map is injective.

A ψ

ε

ψA ψ

1 ε

ψ2A ψ

1 1 ε

. . .

k Pζ P 2

1 ζ . . . P

3.1.5. The “nitary closure”. This construction starts as above, by noting that themap P n +1 → P n , an p → ζ ( p)an , is split by an → an 1. We let P be thedirect limit of the P n ’s along these inclusions.

Then P is also an art algebra. Its decomposition is dened on P n as above:ψ(m p) = m p for m M X , p P n , m p P n +1 .

In some sense, P is the maximal P -art algebra, and P is a minimal P -artalgebra. More precisely:

Proposition 3.2. Let F be an augmented algebra generated by a set S , and let ψ : F →M X (F) be a map such that ψ(F)x,y = F for all x, y X . Set P = εψ(F) ≤M X , and assume that the augmentation : F →k factors to ζ : P →k.

Then there exists a unique art subalgebra A of P that is generated by S and has decomposition map induced by ψ through the canonical map F →A.

Proof. For all n N there exists a map π n = εψn : F → P n , and these maps arecompatible in that (1 n ζ )πn +1 = πn . There is therefore a map π : F →P , andwe let A be the image of π. This proves the existence part.

Let A ′ = F / J ′ be another image of F in P . Write J = ker π. Then by denitionof art algebra the images of A in P n must be πn (F), so J ′ ≤ker πn , and J ′ ≤ J . Itfollows that J ′ = J , because A and A ′ are both contained in P .

If X =

{1, . . . , q

}, then a maximal augmented subalgebra of M X is P = M q

−1 k,

where the augmentation vanishes on M q−1 . The examples we shall consider fall intothis class.

For V a vector space, we denote by V ◦ its dual, and we consider V V ◦ as asubspace of End( V ), under the natural identication ( v ξ )(w) = ξ (w) ·v.

3.1.6. Branchness. Let A be a recurrent transitive algebra. We say that A isweakly branch , if for every v X , writing |v| = n , we have φn (A) ∩(A

(v v◦)) = {0}, where v v◦ is the rank-1 projection on kv ≤kX n ;weakly regular branch , there exists a non-trivial ideal K A, called the branch-

ing ideal , with M X (K ) ≤ψ(K );branch , if for all n N the ideal φn (A) ∩(A (v v◦)) : v X n has nite

codimension in φn (A);regular branch , if there exists a nite-codimension ideal K A with M X (K )

≤ψ(K ).Proposition 3.3. Let A be an art algebra. Then it is innite-dimensional.

If A is regular branch, then it is weakly regular branch and branch; if it is branch,then it is weakly branch. If it is weakly regular branch, then it is weakly branch.

Proof. Let A be an art algebra; then it is unital. By assumption, the map ψx,y :a → ψ(a)x,y is onto. Choose any x = y; then since ψ(1)x,y = 0, so ψx,y is notone-to-one. It follows that A is innite-dimensional.




Let now A be regular branch, with branching ideal K . Since A is innite-dimensional, K = 0, so A is regular weakly branch. Assume now only that K isnon-trivial, and let v

X be any vertex. Since M

Xn (K )

≤ ψn (A), we may take

any a = 0 in K and consider the element ψ−n (a (v v◦)) = 0 in A. This showsthat A is weakly branch. The other implications are of the same nature.

The choice of v in the denition of weakly branch algebra may have seemedarticial; the following more general notion is equivalent:

Lemma 3.4. Let A be a weakly branch algebra. Then for any n N and any ξ, η kX n there exists a = 0 in A with (1 −P ξ )(ψn a) = 0 = ( ψn a)(1 −P η ), where P ξ , P η M X n denote respectively the projectors on ξ, η.

Proof. The weakly branch condition amounts to the lemma for ξ = η a basis vector(element of X n ) of kX n . Write in full generality ξ =

ξ v v and η =

ηv v, the

sums running over v

X n . Fix w

X n and choose b

= 0 with b (w w◦)

ψn (A). For all v, w X n choose cv,w with vψn (cv,w ) = w; this is possible becauseprojection on the ( v, w) entry is a surjective map: A →A. Finally set

a =v,w X n

ξ v cv,v 0 bcv0 ,w ηw .

3.2. Hausdorff dimension. Let A be an art algebra. For every n , it has a rep-resentation πn = ψ n : A → M X n (k). We dene the Hausdorff dimension of Aas

Hdim( A) = liminf n →∞

dim πn (A)dim M X n

.

Let us compute the Hausdorff dimension of the tree closure P dened in 3.1.4.There, πn (P ) is none other than P n . Let n = ker ζ n denote the augmentationideal of P n . Then, as a vector space, P n +1 = M X n P , so

dim P n +1 = dim( M X )(dim P n −1) + dim P .

It follows that

dim P n = dim P −1dim M X −1

(dim M X )n + dim M X −dim P

dim M X −1 ,

and since dim P 0 = 1 we have

Hdim( P ) = dim P −1

# X 2 −1 .

If A is a P -art algebra, we dene its relative Hausdorff dimension as

Hdim P (A) = Hdim( A)Hdim( P )

= Hdim( A)dim P −1# X 2 −1

.

Proposition 3.5. Let A be a regular branch P -art algebra. Then HdimP A is a rational number in (0, 1].




Proof. Let A have branching ideal K , and for all n N set An = πn (A). Let M Nbe large enough so that A/ψ −2M X 2 (K ) maps isomorphically into AM /π M −2(K ).We then have, for all n

≥M ,

dim An = dim( A/ K ) + dim πn (K )

= dim( A/ K ) + dim( ψK /M X (K )) + # X 2 dim πn −1 (K )

= (1 −# X 2)dim( A/ K ) + dim( ψK /M X (K )) + # X 2 dim An −1 .

We write dim An = α# X 2n + β , for some α, β to be determined; we have

α# X 2n + β = (1 −# X 2) dim( A/ K ) + dim( ψK /M X (K )) + # X 2(α# X 2( n −1) + β ),

so β = dim( A/ K ) − dim(ψK /M X (K ))/ (# X 2 − 1). Then set α = (dim AM −β )/ # X 2M . We have solved the recurrence for dim An , and α > 0 because Anhas unbounded dimension, since A is innite-dimensional by Proposition 3.3.

Now it suffices to note that Hdim( A) = α to obtain Hdim P (A) > 0. Further-

more only linear equations with integer coefficients were involved, so Hdim(A

), andHdim P (A), are rational.

3.3. Tree enveloping algebras. Let G be a recurrent group, acting on a tree X .We therefore have a map kG →End( kX ω ), obtained by extending the representa-tion G →Aut X ω by k-linearity to the group algebra. We dene the tree enveloping algebra of G as the image A of the group algebra kG in End( kX ω ).

This notion was introduced, slightly differently, by Said Sidki in [38]; it has alsoappeared implicitly in various places, notably [9] and [29].

Lemma 3.6. Let A be a quotient of the group ring kG, and let H ≤ G be a subgroup. Let K ≤A be the right ideal generated by {h −1 |h H }. Then

dim A/ K ≤[G : H ].

Proof. It clearly suffices to prove the claim for A = kG. Let n = [G : H ] be theindex of H in G, and let T be a right transversal of H in G. Given a A , writea = a(gi )gi and each gi = hi t i for some h i H, t i T . Then we have

a = a(gi )h i t i = a(gi )t i + a(gi )(h i −1)t i ,

so T generates A / K .

Theorem 3.7. Let G be a recurrent transitive group, and let A be its tree enveloping algebra.

(1) A is an art algebra.(2) If G is either a weakly branch group, a regular weakly branch group, a branch

group, or a regular branch group, then A enjoys the corresponding property.

Proof. Let G be a recurrent transitive group, with decomposition φ : G →G S X .Set F = kG. We dene ψ : F →M X (F) by extending φ linearly: for g G, set

ψ(g) =x X

(g@x) (xg x◦).

We also let P be the image of kS X in M X ; since S X is 2-transitive, P = M # X −1 k.By Proposition 3.2 there is a unique image of F that is an art subalgebra of F,

and by construction this image is A .




Assume that G is regular branch, with branching subgroup K . Set

K = k −1 : k K .

Then K is an ideal in A , of nite codimension by Lemma 3.6. Since x K ≤φ(K ),we have K (x x◦) ≤ ψ(K ) for all x X , and since A is transitive we getM X (K ) ≤ψ(K ), so A is regular branch.

Next, assume G is weakly branch, and pick v X n . There exists 1 = g G withg|X ω \vX ω = 1, say g = v h. Then g−1 = 0, and 0 = ψn (g−1) = ( h −1)(v v◦)A (v v◦), proving that A is weakly branch. The other implications are provensimilarly.

We note that the tree enveloping algebra corresponding to the odometer ( §2.4.7)or the lamplighter group ( §2.4.8) are isomorphic to their respective group ring.Indeed these groups have a free orbit in their action on X ω . Branch groups are atthe extreme opposite, as we will see below.

Question 2. If A is the tree enveloping algebra of a branch group G, does Hdim( G) >0 imply Hdim( A) > 0? do we even have HdimP (A) ≥Hdim p(G) for G ≤W p?

3.3.1. Algebraic Properties. Recall that an algebra A is just-innite if A is innite-dimensional, and all proper quotients of A are nite-dimensional (or, equivalently,all non-trivial ideals in A have nite codimension). The core of a right ideal K ≤A isthe maximal 2-sided ideal contained in K . The Jacobson radical R is the intersectionof the maximal right ideals.

An algebra A is prime if, given two non-zero ideals I , J ≤A, we have IJ = 0. Itis primitive if it has a faithful, irreducible module, or equivalently a maximal rightideal with trivial core. It is semisimple 3 if its Jacobson radical is trivial.

Lemma 3.8. Let G be a regular branch group, with branching subgroup K . Let Abe its tree enveloping algebra, with branching ideal K . If either K/ [K, K ] is nite,

or G is nitely generated, then K / K 2

is nite-dimensional.Proof. Consider K = k −1 : k K ≤kG. Then given k1 , k2 K we have

[k1 , k2 ]−1 = k−11 k−1

2 (k1 −1)(k2 −1) −(k2 −1)(k1 −1) K 2 ,

so K 2 contains [ K, K ] −1. This holds a fortiori in A, so if K/ [K, K ] is nite theresult follows from Lemma 3.6.

If G is nitely generated, then A is also nitely generated, so all its nite-codimension subrings are also nitely generated [26]. In particular K / K 2 is nite-dimensional, and if I , J are ideals of nite codimension, then so is IJ .

Theorem 3.9. Let A be a regular branch tree enveloping algebra. Then any ideal J ≤A contains M X n (K 2 ) for some large enough n N.

In particular, if K / K 2 is nite-dimensional, then A is just-innite, and if K 4 = 0 ,

then A is prime.Proof. Assume A is the tree enveloping algebra of the group G. Let J be a non-trivial ideal of A, and chose any non-zero a J . Then a = a(gi )gi , and thenitely many gi in the support of a all act differently on X . The entries of ψn (a),for large enough n , are therefore monomial; more precisely, there exist v, w X nsuch that the ( v, w) entry of ψn (a), call it b, is in G, and therefore is invertible.

3aka semiprimitive or J -semisimple




Since J is an ideal, we have for any v′, w′ X n

(K (v′ v◦))a(K (w (w′)◦)) = ( K bK ) (v′ (w′)◦)

≤ J .

It follows that J contains M X n (K 2 ), which by assumption is conite-dimensional.Assume now that J , J ′ are two non-zero ideals of A. By the above, there are

n, n ′ N such that J contains M X n (K 2) and J ′ contains M X n ′ (K 2 ). For m largerthan n, n ′ we then have 0 = M X m (K 4 ) ≤ JJ ′.

Recall also that an algebra A is PI (“Polynomial Identity”) if there exists w = 0in the free asssociative algebra k{v1 , . . . , vk} such that w(a1 , . . . , a k ) = 0 for alla i A. The following result is analogous to 2.5:

Theorem 3.10. Let A be a weakly branch art algebra. Then it is not PI.

We prove the theorem using the following result, which may be of independent

interest. Let A be an algebra acting faithfully on a vector space V . We say thatA separates V if for every nite-dimensional subspace Y of V and any ξ Y thereexists a A with Y a = 0 and ξa Y, ξ .

Proposition 3.11. Let A be an algebra separating a vector space V . Then A is not PI.

Proof. Let P k{v1 , . . . , vk} be a non-commutative polynomial. We will nda1 , . . . , a k A and η V such that ηP (a1 , . . . , a k ) = 0. We actually will showmore, by induction: let X 0 {v1 , . . . , vk} be the set of monomials, without theircoefficients, appearing in P , and let X be the set of prexes of words in X 0 . Forany η = 0 V , we construct ( a1 , . . . , a k ) Ak such that {ηx(a) |x X } is anindependent family. It then of course follows that ηP (a) = 0.

The induction starts with X =

{1

}. Then any η

= 0 will do. Let now X

contain at least two elements, and let y = v p . . . vqvr be a longest element of X .By induction, there exists a Ak such that Y 0 = {ηx(a) |x X \ {y}} is anindependent family. If ηy(a) is linearly independent from Y 0 , we have nothing todo. Otherwise, take ξ = η(v p . . . vq)(a) and Y = Y 0 \ {ξ }. Since V is separatedby A, there exists b A with Y b = 0 and ξb Y, ξ . Set a′i = a i for i = r , anda ′r = ar + b. Then {ηx(a ′) |x X } is an independent family.

Proof of Theorem 3.10. The algebra A is a subalgebra of P , which by denitionis a subalgebra of lim

←−M n

X . We may therefore assume that A is a subalgebra of End( V ) for the vector space V = lim

←−kX n .

Let Y be a nite-dimensional subspace of V , and let ξ V be any. Let πn

be the projection V → kX n . Since Y is a closed subspace, there exists n Nsuch that v = π

n(ξ ) π

n(Y ), and furthermore such that there is also w π

n(V )linearly independent from v and πn (Y ). By Lemma 3.4 there exists a A which

annihilates Y while it sends v to a multiple of w. Consider all possible such a; if they all annihilated ξ , then they would also annihilate the orbit of ξ under P u A,where P u M X n denotes projection on u; since they also annihilate V (1 −P u ),they would all annihilate V , whence a = 0 because the representation V is assumedfaithful. This contradicts the condition that A is weakly branch.

We may therefore apply Proposition 3.11 to conclude that A is not PI.




3.3.2. Compatible ltrations. Let A be the tree enveloping algebra of a regularbranch group G. We have three descending ltrations of A by ideals, namelypowers of the branching ideal ( K n ); powers of the augmentation ideal ( n ); and(M X n (K )).

Proposition 3.12. Assume that there is an n N such that M X n (K ) is contained in K 2 . Then the normal subgroups of G control the ideals of A: given any non-zeroideal J ≤A, there exists a non-trivial normal subgroup H G with H −1 J .

Proof. By Theorem 3.9, there is n N such that J contains M X n − 1 (K 2), so containsM X n (K ). Set H = φ−n (K X

n); then J contains H −1.

Corollary 3.13. Assume that there is an n N such that M X n (K ) is contained in K 2 . Then A is just-innite and prime.

Proposition 3.12 may be used to obtain some information on the Jacobson radicalof A :

Proposition 3.14 ([38, Corollary 4.4.3]) . Assume that normal subgroups of Gcontrol ideals of A; that G is just-innite- p (i.e. that all proper quotients of G are nite p-groups), and that k is of characteristic p. Then either R = 0 or R = .

Proof. R ≤ since is a maximal right ideal. If R = 0, then there is a non-trivialH G with H −1 R . Since G/H is a nite p-group, A / R is a nilpotent algebra,so is 0, and R = .

This in turn gives control on representations of A , by the following result:

Proposition 3.15 ([16]). Let A be a just-innite, semiprimitive, nitely generated k-algebra over an uncountable eld k. Then either A is primitive, or A satises a polynomial identity.

Since weakly branch art algebras satisfy no polynomial identity (Theorem 3.10),they admit irreducible faithful representations as soon as they are semiprimitive.

3.3.3. The tree enveloping algebra of P . Consider as in §2.2.1 a subgroup P of S X ,and its tree closure P ≤ Aut (X ). It is regular branch, with branching subgroupP .

Proposition 3.16. Let A be the tree enveloping algebra of P , and let P be the image in M X of kP . Then A = P .

Proof. Since A ≤P , it suffices to show that the natural map kP →P n is onto forevery n. Let denote the augmentation ideal of kP ; then

ψ(kP ) = M X ( ) + 1 P ,

and therefore ψn (kP ) = M X n ( ) + 1 P n , and the result follows.

The algebra P can be dened in a different way, following [38]. The group P isa pronite (compact, totally disconnected) group, and therefore kP is a topologicalring. Consider the ideal

(8) J = (v g −1)(w h −1) : v = w X n for some n; g , h P

in kP . On the one hand, J has trivial image in P , since in ψn (v g−1) and ψn (wh −1) are diagonal matrices with a single non-zero entry, in different coordinatesv, w. On the other hand, all relations in the matrix ring M X n (kP ) can be reduced




to these. It follows that P equals kP / J , where J , the “thinning ideal”, denotes theclosure4 of J in the topological ring kP . More details appear in §4.1.

For any recurrent group G, we may now consider G as a subgroup of some P ,and therefore kG is a subalgebra of kP . The tree enveloping ring of kG is thenkG/ (kG ∩ J ). This was the original denition of tree enveloping rings.

3.4. Lie algebras. In this subsection, we let p be a prime, k = F p, and x X =

{1, . . . , p}. Let G be a recurrent subgroup of W p, with decomposition φ : G →G C pwhere C p is the cyclic subgroup of S X generated by (1 , 2, . . . , p ). We dene thedimension series (Gn ) of G by G1 = G, and

Gn = g G : either g = [h, k ] for h Gn −1 , k G or g = h p for h G n/p .

Since G is residually- p, we have Gn = {1}.The quotient Gn /G n +1 is an F p-vector space, and we form the “graded group”

gr G =

n ≥1

Gn /G n +1 .

Multiplication and commutation in G endows gr G with the structure of a gradedLie algebra over F p, and x →x p induces a Frobenius map on gr G, turning it intoa restricted Lie algebra.

The dimension series of G can be alternately described, using the augmentationideal of F pG, as

Gn = {g G |g −1 n }.Furthermore, consider the graded algebra gr F pG = n ≥0

n/

n +1 associated tothe descending ltration ( n ) of F pG. Then

Proposition 3.17 (Quillen [36]) . gr F pG is the restricted enveloping algebra of gr G.

3.4.1. Graded tree enveloping algebras. Let A be the tree enveloping algebra of theregular branch group G, and assume that A is a graded algebra with respect to theltration ( n ). Then gr G embeds isomorphically in A .

Proposition 3.18. Assume that A is a quotient of gr F pG. Then the natural mapgr G →gr F pG induces an embedding gr G →A.

Proof. Let a gr G be such that its image in A is trivial. Then, since A is graded, allthe homogeneous components of a are trivial. But these homogeneous componentsbelong to quotients Gn /G n +1 along the dimension series of G, and since G → A ,they must be trivial in Gn /G n +1 . We deduce a = 0.

If we forget for a moment the distinction between kG and gr kG, Proposition 3.18can be made more conceptual, by returning to the “thinning process” describedafter (8): assume G factors as A

×B . Then

kG =

kA

kB , and the “thinning”

process maps kG to

kG/ J = ( kA kB )/ {(1, 0) = (0 , 1)},

with J = (kA) (kB ). We have gr A kA and kB kB and gr G = gr Agr B kG/ J . It is in this sense that thinning “respects” Lie elements. More detailsare given in §4.1.

4note that [38] does not mention this closure, although it is essential




Proposition 3.18 applies in particular to the group P and its tree envelopingalgebra P . This points out the recursive structure of gr P , as described in [12].

3.5. Gelfand-Kirillov dimension. Let A be an algebra (not necessarily associa-tive), with an ascending ltration ( Fn ). Assume F−1 = 0, and Fn / Fn −1 is nite-dimensional for all n N. Then the Hilbert-Poincare series of A is the formal powerseries

ΦA (t) =∞

n =0an tn =

n ≥0

dim( Fn / Fn −1)tn .

In particular, if A is generated by a nite set S , it has a standard ltration denedas follows: Fn is the linear span of all at-most- n-fold products s1 . . . s k for all k ≤n,in any order (if A is not associative).

If A = n ≥0 An is graded, we naturally lter A by setting Fn = A 0 + · · ·+ An .If the coefficients an grow polynomially, i.e. an ≤ p(n) for some polynomial p,

then A has polynomial growth , and its (lower) Gelfand-Kirillov dimension is denedas

GKdim( A) = liminf n →∞

log an

log n .

If A is nitely generated and Fn is the span of at-most- n-fold products of generators,then this limit does not depend on the choice of nite generating set.

If A is nitely generated and either Lie or associative, then the coefficients anmay not grow faster than exponentially. A wide variety of intermediate types of growth patterns have been studied by Victor Petrogradsky [33, 34].

Let G be a group, with Lie algebra gr G. Then the Poincare-Birkhoff-WittTheorem gives a basis of gr F pG consisting of monomials over a basis of gr G, withexponents at most p −1. As a consequence, we have the

Proposition 3.19 (Jennings [24]) . Let G be a group with dimension series (Gn ),

and set ℓn = dim F p (Gn /G n +1 ). Then

Φgr F p G (t) =∞

n =1

1 −t pn

1 −tn

ℓn

.

Approximations from analytical number theory [27] and complex analysis givethen the

Proposition 3.20 ([35], Theorem 2.1) . With the notation above for ℓn , and an =dim n

/n +1 , we have

(1) {an } grows exponentially if and only if {ℓn } does, and we have

limsupn →∞

ln ℓn

n = limsup

n →∞ln an

n .

(2) If ℓn nd , then an en ( d +1) / ( d +2) .

A lower bound on the growth of a group G may be obtained from the growth of F pG:

Proposition 3.21 ([20], Lemma 8). Let G be a group generated by a nite set S ,and let f (n) be its growth function. Then

f (n) ≥dim( n/

n +1 ) for all n N.




It follows that if gr G has Gelfand-Kirillov dimension d, then G has growth at least exp(n (d+1) / (d+2) ).

It follows that a non-nilpotent residually- p group has growth at least exp( √ n).It also follows that 1-relator groups that are not virtually abelian have exponentialgrowth [15].

Theorem 3.22. Let G be a contracting group in the sense of §2.3.1, acting on the tree X . Let A be its tree enveloping algebra. Then A has Gelfand-Kirillov dimension

(9) GKdim( A) ≤2nlog# X

−log λ;

in particular, if (X ω , ) has polynomial growth of degree d, then A has Gelfand-Kirillov dimension at most 2d.

Proof. Let S be the chosen generating set of G, and write f (r ) = dim k (kS r ). Then

by contraction kS r M X n (kS λr + K ),so f (r ) ≤ # X 2n f (λr + K ). It follows that log f (r )/ log r converges to the valueclaimed in (9).

The last remark follows immediately from Proposition 2.7.

Question 3. Assume furthermore that G is branch. Do we then have equality in (9)?

4. Examples of Tree Enveloping Algebras

We describe here in more detail some tree enveloping algebras. Most of theresults we obtain concern Grigorchuk’s group. They are modeled on the followingresult. Said Sidki considers in [38] the tree enveloping algebra A of the Gupta-Sidkigroup Γ of §2.4.3, over the eld F3 . He shows:

Theorem 4.1. (1) The group Γ and the polynomial ring F3[t] embed in A;(2) The algebra A is just-innite, prime, and primitive.

4.1. The “thinning process”. We recall and generalize the original constructionof A, since it is relevant to §3.4.1. Let G → G P be a recurrent group, withP ≤S X . Let F = kG be its group algebra. Then we have a natural map

F →F X P = F X kP,

where A P designates the crossed product algebra; the indicates the tensorproduct as vector spaces, with multiplication

(1 X π)(g1 · · · gq 1) = ( g1π · · · gqπ 1) (1 X π)

for all g1 , . . . , gq G and π P .We wish to construct a quotient of F which still contains a copy of G. For this,

let i denote, for all i X , the augmentation ideal of the subalgebra k · · · F

· · · k = F , with the ‘ F’ in position i; and let I i denote the ideal in kP generatedby {π −1 | iπ = i}. Set then

J =i= j X

i j kP +i X

I i +i X

k I i .




Lemma 4.2 ([38, §3.2]). F/ J = M X (F).

This process can then be iterated, by thinning the ‘ F’ on the right-hand side of the above; the limit coincides with the tree enveloping ring of G.

4.2. Grigorchuk’s group. From now on, we restrict to the Grigorchuk group Gdened in §2.4.2. There are two main cases to consider, depending on the charac-teristic of k: tame ( = 2) or wild (= 2).

We begin by some general considerations. As generating set of G we alwayschoose S = {a,b,c,d}, and as generating set of its tree enveloping algebra A wemay choose S = {A,B,C,D }, dened by A = a−1, B = b−1, C = c−1, D = d−1.Therefore the augmentation ideal of A is generated by S .

Since G’s decomposition is G →G C 2 , we have

P = a bb a a, b k = k[Z/ 2].

Theorem 4.3. The algebra A is regular branch, just-innite, and prime.

Proof. A is regular branch by Theorem 3.7. By Lemma 3.8 and Theorem 3.9 it is just-innite and prime.

Ana Cristina Vieira proved in [41, Corollary 4] that A is just-innite if k = F2 .Actually her arguments extend to arbitrary characteristic, and also show that A isprime.

4.2.1. Characteristic = 2 . If k is an uncountable eld of characteristic = 2, thenby [4,31] the algebra A is semisimple. However, the question of semisimplicity of QG or F pG for p = 2 seems to be open.

Let now k be any eld of characteristic

= 2.

Proposition 4.4. The algebra A has relative Hausdorff dimension HdimP (A) = 1 .

Proof. This is a reformulation of [9, Theorem 9.7], where the structure of the nitequotient πn (A) is determined for k = C. The result obtained was

πn (A) = C +n −1

i=0

M 2i .

It follows that πn (A) has dimension (4 n + 2) / 3. The proof carries to arbitrary k of characteristic = 2.

The algebra A does not seem to have any natural grading; indeed if denotethe augmentation ideal of A , then

2 = , because is generated by idempotents12 (1 −a), 1

2 (1 −b), 12 (1 −c), 1

2 (1 −d). As a side note, the Lie powers [n ] of ,dened by

[1] = and

[n +1] = A ab −ba a [n ], b A,

also seem to stabilize.The following presentation is built upon Proposition 2.9. Since the proof is

similar to that of Theorem 4.14, we only sketch the proof.




Theorem 4.5. Consider the endomorphism σ of k{a,b,c,d} dened on its basis by

(10) a →aca, b →d, d →c, c →band extended by linearity. Then

(11) A = a,b,c,d a2 = b2 = c2 = d2 = bcd = 1 ,

σn (d −1)a(d −1) = σn (d −1)a(dacac −1) = 0 n ≥0 .

Proof. Let F be the free associative algebra on S ; dene ψ : F → M 2(F ) usingformulæ (12). Set J 0 = a2 −1, b2 −1, c2 −1, d2 −1,bcd−1 , J n +1 = ψ−1(M 2( J n )),and J = n ≥0 J n . We therefore have an algebra A′ = F/ J , with a natural mapπ : A ′ →A which is onto. We show that it is also one-to-one.

Take x ker π. Then it is a nite linear combination of words in S , so thereexists n N such that all entries in ψn (x) are linear combinations of words of syllable length at most 1, where a’s and

{b,c,d

}’s are grouped in syllables. Since

they must also act trivially on kX ω , they belong to J 0 ; so x J n .It remains to compute J n . First, J 1 / J 0 is generated by all ( du −1)a(dv −1)

for u, v {a,b,c,d} with an even number of a’s. It is sufficient to consider onlyu = 1; and to assume that v contains only a’s and c’s; indeed d’s can be pulled outto give a shorter relator of the form ( d −1)a(dw −1), and b’s can be replaced byc’s by the same argument. Using the previous relators, we may then suppose thatv is of the form (ac)2k .

Next, the relators rk = ( d −1)a(d(ac ) 2 k

−1) J 1 lift to generators σn (r k ) of J n +1 / J n .

Finally, using the relator σ(r 0 ) = cacac −aca , we see that it is sufficient toconsider the relators σn (r 0) and σn (r 1).

Although we may not grade A , we may still lter it by powers of the generatingset S . We give the following result with minimal proof; it follows from argumentssimilar, but harder, than those in Proposition 4.16.

Theorem 4.6. The algebra A has Gelfand-Kirillov dimension 2.More precisely, set Fn = n

i =0 kS i and an = dim F n / F n −1 . Then a1 = 4 , a2 =6, a 3 = 8 , a 4 = 10 , a 5 = 13 , a 6 = 16 , and for n ≥7

an =

4n − 32 2k if 2k ≤n ≤ 5

4 2k ,3n − 1

4 2k if 54 2k ≤n ≤ 3

2 2k ,n + 11

4 2k if 32 2k ≤n ≤ 7

4 2k ,2n + 2 k if 7

4 2k ≤n ≤2k+1 .

It follows for example that, if n is a power of two greater than 4, then

dim Fn = 43

n 2 + 54

n + 23

.

Note that A has Gelfand-Kirillov dimension at most 2, by Theorem 3.22; fur-thermore, it cannot have dimension 1 since G satises no identity, so by Bergman’sgap theorem [25] it has dimension 2.

Lemma 4.7. Set x = ab −ba and let K = AxA be the branching ideal of A. Then A/ K is 6-dimensional, and K /M 2 (K ) is 20-dimensional.




Proof. The codimension of K is at most 16, which is the index of K in G. We thencheck y = (1+ b)(1−d) K , because y = 1

2 (c−1)xay , and we use (d−1)a(d−1) = 0to see that the codimension of K is at most 6, with transversal

{1,a,d,ad,da,ada

}.

These elements are easily seen to be independent modulo K .The assertion on K /M 2(K ) has a similar proof.

Sketch of the proof of Theorem 4.6. The rst few values of an are computed di-rectly. We consider the ltrations En = Fn ∩K and D n = Fn ∩M 2(K ) of K andM 2(K ) respectively. For n ≥ 3, we have dim F n / En = 6, and for n ≥ 6 we havedim En / D n = 20. It follows that an = dim D n / D n −1 for n large enough, and weplace ourselves in that situation.

If X is a subspace of A generated by words in {a,b,c,d} , we let X at denotethe subspace spanned by words starting in a and ending in {b,c,d}; we denesimilarly X aa , X ta and X tt . We x X n = kS n . Using contraction of G and theendomorphism (10), we then show that for n ≥3

X 4n = X 4nat + X 4nta = X 2nta 00 X 2n

at+ X

2nat 00 X 2n

ta= X

2n

00 X 2n ,

X 4n +1 = X 4n +1aa + X 4n +1

tt = 0 X 2nta

X 2nat 0 + X 2n +1

aa 00 X 2n +1

tt,

X 4n +2 = X 4n +2at + X 4n +2

ta = 0 X 2n +1aa

X 2n +1tt 0 + 0 X 2n +1

ttX 2n

aa 0 = 0 X 2n +1

X 2n +1 0 ,

X 4n +3 = X 4n +3aa + X 4n +3

tt = X 2n +1tt 00 X 2n +1

aa+ 0 X 2n +2

atX 2n +2

ta 0 ;

we therefore havea2n = 2an , a2n +1 = an + an +1 ,

from which the claim follows.

4.2.2. k = F2 . If we let k be of characteristic 2, then sharper results appear. Werst recall, in a more concrete form, the results stated above for general k.

Proposition 4.8. The algebra A is recurrent; its decomposition map ψ : A →M 2(A) is given by

(12) A →1 11 1 , B →

A 00 C , C →

A 00 D , D →

0 00 B .

Proof. The expression of ψ follows from the denition. Upon inspection, one sees 1,B , C and D in the (2 , 2) corner as ψ(A), ψ(D ), ψ(B ) and ψ(C ); then ψ(ACA + C )gives A a B in the (2 , 2) corner, so projection on the (2 , 2) corner is onto. For theother corners, it suffices to multiply the above expressions by 1 + A on the left,on the right, or on both sides to obtain all generators in the image of the ( i, j )

projection. Theorem 4.9. The relative Hausdorff dimension of A is HdimP (A) = 7 / 8.

Proof. Let A n be the nite quotient πn (A) of A , and set bn = dim An . Then b2 = 8by direct examination, and one solves the recurrence, for n ≥3,

bn +1 = dim An = dim A/ K + dim πn +1 (K )

= 6 + dim( K /M 2 (K )) + 2 2 dim πn (K ) = 6 + 8 + 4( bn −6)




to bn = (14 ·4n −2 +10) / 3. This gives Hdim( A) = 14 / 24, and Hdim P (A) = 7 / 8.

Let H be the stabilizer in G of the innite ray 1 ω X ω ; then by [9] it is a weaklymaximal subgroup, i.e. if H I ≤ G then I has nite index in G. It follows thatthe right ideal J = ( H −1)A is a “weakly maximal” right ideal, i.e. if J I ≤ Athen I has nite codimension in A. Since the core of J is trivial, it follows that Aadmits a faithful module A/ J all of whose quotients are nite. This is none otherthan the original representation on kX .

Proposition 4.10. The ideal J has Gelfand-Kirillov dimension 1; i.e. the dimen-sions of the quotients J ∩ n J ∩ n +1 are bounded.

Proof. This is a reformulation of [7, Lemma 5.2], where the uniseriality of themodules naturally associated with X m is proven.

From now on, we identify A with its image in M 2(A). We also commit the usualcrime of identifying words over S with their corresponding elements in A . Set

(13) R0 = {A2 , B 2 , C 2 , D 2 , B + C + D,BC,CB,BD,DB,CD,DC,DAD }.

We also set T = {B,C,D }.

Lemma 4.11. All words in R0 are trivial in A. Furthermore, the last relator is part of a more general pattern: DwD is trivial for any word w S with |w| ≡ 1mod 4.

Proof. Clearly A2 = 0. Then B + C + D = ( 0 00 B + C + D ) so B + C + D acts trivially

on kX ω and is therefore trivial. Given any x, y T we have xy = ( 0 00 x ′ y ′ ) for some

x′, y′ T and these are therefore also relations. Finally, let w S be a word of length 4 n + 1. Clearly, by the above, DwD = 0 unless possibly if w is of the formAx1 . . . Ax 2n A for some xi T . Then w = ( w11 w12

w21 w21 ) where each wij is a linearcombination of words that either start or end in T ; multiplying on both sides with

D = 0 00 B therefore annihilates DwD .

4.2.3. A recursive presentation for A. Consider the substitution σ : S → S ,dened as follows:

A →ACA, B →D, C →B, D →C.

We say that a word w S is an A ÷T word if its rst letter is A and its last letteris in T ; we dene similarly A ÷A, T ÷A and T ÷T words.

Lemma 4.12. Let w S represent an element of K . Then in A we have

• if w is a A ÷A word, then σ(w) = w ww w ;

• if w is a A ÷T word, then σ(w) =0 w0 w ;

• if w is a T ÷A word, then σ(w) = 0 0w w ;

• if w is a T ÷ T word, then σ(w) = 0 00 w , unless if w belongs to

{CAC,CAD,DAC,DAD }, in which case σ(w) = ADA 00 w .




Note in particular that because of the four exceptional cases for T ÷T words, themap σ does not induce an endomorphism of A . It seems that there does not exista graded endomorphism τ of A with ψ(τ (w))

2,2 = w for all long enough w S .

Proof. The induction starts with the words B, CAC, CAD, DAC,DAD . If for ex-ample w is a A ÷T word, we have σ(w) = ( 0 w

0 w ), and therefore

σ(wA) = 0 w0 w ACA = 0 w

0 wA + D A + DA + D A + D = wA wA

wA wA ,

where wD = 0 because w ends in a letter in T .

Proposition 4.13. The algebra A is regular branch.

Proof. This follows from Theorem 3.7. Alternatively, consider the ideal

K = ADA,AB,BA .

Compute dim( A/ K ) = 6, with A = K 1,A,B,D,AD,DA . Next check

ADA 00 0 = CACAC = C (ADA )C + CA (BA )C K

AB 00 0 = CADA = C (ADA ) K

BA 00 0 = ADAC = ( ADA )C K ,

giving M 2(K ) ≤K . We have dim K /M 2(K ) = 8, becauseK = M 2(K ) ADA,AB,BA,ABA,BAB,ABAB,BABA,ABABA .

We may also easily check that K / K 2 is 12-dimensional, by

K 2 = K AB,BA,ABA,ADA,BAB,BAD,DAB,

ABAD, ADAB, BADA, DABA, DABAD .

Theorem 4.14. Let R0 be as in (13). Then the algebra A admits the presentation A = A,B,C,D |R0 , σn (CACACAC ), σn (DACACAD ) for all n ≥0 .

Corollary 4.15. A is graded along powers of its augmentation ideal . This grading coincides with that dened by the generating set S .

Proof. All relations of A are homogeneous — they are even all monomial, exceptfor B + C + D .

Proof of Theorem 4.14. Let F be the free associative algebra on S ; dene ψ : F →M 2(F) using formulæ (12). Set J 0 = R0 , J n +1 = ψ−1(M 2( J n )), and J = n ≥0 J n .

We therefore have an algebra A′ = F / J , with a natural map π : A ′ → A which isonto. We show that it is also one-to-one.

Take x ker π. Then it is a nite linear combination of words in S , so thereexists n N such that all entries in ψn (a) are words in A or T . Since they mustalso act trivially on kX ω , they belong to J 0; so x J n .

It remains to compute J n . First, J 1/ J 0 is generated by all DwD with |w| ≡ 1mod 4, which map to 0 F / J 0 , and CACACAC , which maps to DAD = 0 F / J 0 .Using the relation r0 = DAD , we see that all DwD are consequences of r1 =




DACACAD and r2 = CACACAC . For example, DACABAD = r 1 + DACAr 0 ,r ′1 = DABABAD = DACABAD + r 0ABAD , and for n ≥2, by induction

r ′n = D(AB )2n AD = r ′n −1 ABAD + r ′n −2 ABACABAD+ D (AB )2n −4 A CABACAr 0 + CAr 1 + r2 AD

Finally, the relations r1 , r 2 J 1 lift to generators σn (r 1), σn (r2 ) of J n +1 / J n .

Proposition 4.16. Successive powers of the augmentation ideal of A satisfy, for n ≥3,

dim( n/

n +1 ) =2n − 1

2 2k if 2k ≤n ≤ 32 2k

n + 2 k if 32 2k ≤n ≤2k+1 .

It follows that, although kG has large growth, namely dim( n/

n +1 ) exp(√ n)in kG by Proposition 3.21, the growth of its quotient A is polynomial of degree 2:

Corollary 4.17. A has Gelfand-Kirillov dimension 2, both as a graded algebra (along powers of ), and as a nitely generated ltered algebra.

Proof of Proposition 4.16. Assume n ≥3. Then we have

2n = n 0 10 1 , n 0 0

1 1 ,(14)

2n +1 = n 1 11 1 , n +1 0 0

0 1 .(15)

Indeed consider a generator w S of 2n . Then w is a word of length 2n, so iseither a A ÷T word or a T ÷A word. It follows that ψ(w) = ( 0 u

0 u ) or ( 0 0u u ) for

some u S n , and the ‘ ’ inclusion is shown.Conversely, take u S n ; if the length of u is even, then u is either a T ÷A word

or a A

÷T word, and set w = σ(u). If

|u

| is odd, then u is either a T

÷T word,

and consider w = σ(u)A and Aσ (u), or it is a A ÷A word, and set w′ = σ(u) andw = w′ with its rst or last letter removed. In all cases, w is a word of length2n, and ψ(w) = ( 0 u

0 u ) or ( 0 0u u ), which shows the ‘ ’ inclusion. A similar argument

applies to (15).Set an = dim( n

/n +1 ). Then it is easy to compute

A/ = 1 giving a0 = 1

/2 = A,B,D giving a1 = 3

2/

3 = AB,BA,AD,DA giving a2 = 43/

4 = ABA,ADA,BAB,BAD,DAB giving a3 = 54/

5 = ABAB, ABAD, ADAB, BABA, BADA, DABA giving a4 = 65/

6= ABABA, ABADA, ADABA, BABAB,

BABAD, BADAB,DABAB,DABAD giving a5 = 8 ,

and formulæ (14,15) give

a2n = 2an , a2n +1 = an + an +1 ,

from which the claim follows.

We now show that the ltrations of A by (ωn ), (K n ) and ( M X n (K )) are equivalent:




Proposition 4.18. For all n N we have 3n ≤K n ≤ 2n ,

3·2n ≤M X n (K ) ≤ 2·2n .

Proof. To check the rst assertion, it suffices to note that all non-trivial words of length 3 in S , namely ( AB )A, ADA, (BA )B, (BA )D, D (AB ), belong to K , while allgenerators of K lie in

3 .To check the third inclusion, take w S 3·2n

; then ψn (w) M X n ( 3). Tocheck the fourth inclusion, take a generator w of K , and consider v = σn (w). Since

|w| ≥2, we have |v| ≥2 ·2n so v 2·2n.

4.2.4. Laurent polynomials in A . It may seem, since A has Gelfand-Kirillov dimen-sion 2, that G contains “most” of the units of A . However, G has innite index inA×, and contains an element of innite order:

Theorem 4.19. A contains the Laurent polynomials k[X, X −1].

Proof. Consider the element X = 1 + A + B + AD . It is invertible, with

X −1 = (1 + B )(1 + AC )(1 + ACAC )(1 + A).

Now to show that X is transcendental, it suffices to show that X has innite order;indeed if X were algebraic, it would generate a nite extension of a nite eld, andtherefore a nite ring; so X would have nite order.

Among words w {A,B,AD } , consider the set W of those of the form

w = ( AB )i 1 AD (AB )i 2 AD. . . (AB )i ℓ .

These are precisely the words starting by an A, and ending by a B or a D . Denetheir length and weight as

|w| =

ℓ

j =1 (2i j + 2) , w =

ℓ

j =1 (2i j + 1) .

Consider the words wn dened iteratively as follows: w1 = ADAB , and wn =τ (wn −1) where τ is the substitution τ (AB ) = ( ADAB )3 (AB )2 , τ (AD ) = ( ADAB )4 .Then

ψ3(wn ) = wn −10 10 1

1 10 0

1 01 0 .

Dene σ(n) = 22·8n −17 . Then |wn | = 4 ·8n and wn = σ(n); and wn is the

unique summand of X σ (n ) in W that belongs to 4·8n

. This proves that all powersof X are distinct.

Note that Georgi Genov and Plamen Siderov show in [17] that (1 + A)(B + C ),(1 + A)(B + D) and (1 + A)(C + D) have innite order in the group ring of G.

However, they project to nil-elements in A .Evidently 1 + X belongs to the augmentation ideal , and is also transcendental

— in particular, it is not nilpotent. However, contains many nilpotent elements:

Proposition 4.20 ([41, Theorem 2]) . The semigroup {A,B,C,D } is nil of degree 8.

An seemingly important question, asked in [41], and which I have been unableto answer, is the following:




Question 4. Does A have a trivial Jacobson radical?

It follows from 3.14 that if the Jacobson radical R is not trivial, then R = .It would therefore suffice to exhibit a non-invertible element in 1 + to obtainR = 0. A possibly easier task would be to show that A is graded nil, i.e. thatits non-constant homogeneous elements are nil. This is a weaker statement than“R = 0”: if X is homogeneous and non-nil, then 1 + X is non-invertible. I havenot been able to nd such X .

Bilbo Baggins, Bag End, Hobbiton, The Shire

Laurent Bartholdi, Ecole Polytechnique F ed erale, SB/IGAT/MAD, B atiment BCH,1015 Lausanne, Switzerland

E-mail address : [email protected]

References[1] Alexander G. Abercrombie, Subgroups and subrings of pronite rings , Math. Proc. Cambridge

Philos. Soc. 116 (1994), 209–222.[2] Miklos Abert, Group laws and free subgroups in topological groups , arXiv:math.GR/0306364.[3] Miklos Abert and B´ alint Vir´ ag, Dimension and randomness in groups acting on rooted trees ,

arXiv:math.GR/0212191.[4] Shimson Avraham Amitsur, On the semi-simplicity of group algebras , Michigan Math. J. 6

(1959), 251–253. MR 21 #7256[5] Bilbo Baggins, There and Back Again. . . A Hobbit’s Tale by Bilbo Baggins , in preparation.[6] Yiftach Barnea and Aner Shalev, Hausdorff dimension, pro- p groups, and Kac-Moody alge-

bras , Trans. Amer. Math. Soc. 349 (1997), 5073–5091.[7] Laurent Bartholdi and Rostislav I. Grigorchuk, Lie methods in growth of groups and groups

of nite width , Computational and Geometric Aspects of Modern Algebra (M. et al., ed.),London Math. Soc. Lect. Note Ser., vol. 275, Cambridge Univ. Press, Cambridge, 2000, pp. 1–27.

[8] , On the spectrum of Hecke type operators related to some fractal groups , Trudy Mat.Inst. Steklov. 231 (2000), math.GR/9910102, 5–45.

[9] , On parabolic subgroups and Hecke algebras of some fractal groups , Serdica Math. J.28 (2002), math.GR/9911206, 47–90.[10] Laurent Bartholdi, A Wilson group of non-uniformly exponential growth , C. R. Acad. Sci.

Paris Ser. I Math. 336 (2003), 549–554.[11] Laurent Bartholdi, Rostislav I. Grigorchuk, and Volodymyr V. Nekrashevych, From fractal

groups to fractal sets , Fractals in Graz (P. Grabner and W. Woess, eds.), Trends in Mathe-matics, Birkha¨ user Verlag, Basel, 2003, pp. 25–118.

[12] Laurent Bartholdi, Lie algebras and growth in branch groups , 2004, to appear in Pacic J.Math.

[13] Laurent Bartholdi and B´ alint Vir´ ag, Amenability via random walks , 2003, submitted.[14] Andrew M. Brunner, Said N. Sidki, and Ana Cristina Vieira, A just nonsolvable torsion-free

group dened on the binary tree , J. Algebra 211 (1999), 99–114.[15] Tullio G. Ceccherini-Silberstein and Rostislav I. Grigorchuk, Amenability and growth of one-

relator groups , Enseign. Math. (2) 43 (1997), 337–354.[16] Daniel R. Farkas and Lance W. Small, Algebras which are nearly nite dimensional and their

identities , Israel J. Math. 127 (2002), 245–251. MR1900701 (2003c:16033)[17] Georgi K. Genov and Plamen N. Siderov, Some properties of the Grigorchuk group and

its group algebra over a eld of characteristic two , Serdica 15 (1989), 309–326 (1990).MR1054623 (91h:20006) (Russian)

[18] Rostislav I. Grigorchuk, On Burnside’s problem on periodic groups , Funkcional. Anal. iPriloжen. 14 (1980), 53–54, English translation: Functional Anal. Appl. 14 (1980), 41–43.

[19] , On the Milnor problem of group growth , Dokl. Akad. Nauk SSSR 27 1 (1983), 30–33.[20] , On the Hilbert-Poincare series of graded algebras that are associated with groups ,

Mat. Sb. 180 (1989), 207–225, 304, English translation: Math. USSR-Sb. 66 (1990), no. 1,211–229.




[21] Rostislav I. Grigorchuk and Andrzej Zuk, On a torsion-free weakly branch group dened by a three state automaton , Internat. J. Algebra Comput. 12 (2002), 223–246, International Con-ference on Geometric and Combinatorial Methods in Group Theory and Semigroup Theory

(Lincoln, NE, 2000).[22] Mikhael Gromov, Groups of polynomial growth and expanding maps , Inst. Hautes Etudes

Sci. Publ. Math. (1981), 53–73.[23] Narain D. Gupta and Said N. Sidki, On the Burnside problem for periodic groups , Math. Z.

182 (1983), 385–388.[24] Stephen A. Jennings, The structure of the group ring of a p-group over a modular eld ,

Trans. Amer. Math. Soc. 50 (1941), 175–185.[25] Gunter R. Krause and Thomas H. Lenagan, Growth of algebras and Gelfand-Kirillov dimen-

sion , Revised edition, Graduate Studies in Mathematics, vol. 22, American MathematicalSociety, Providence, RI, 2000, ISBN 0-8218-0859-1.

[26] Jacques Lewin, Subrings of nite index in nitely generated rings , J. Algebra 5 (1967), 84–88. MR 34 #196

[27] Wo Li, Number theory with applications , World Scientic, 1996.[28] Igor G. Lysionok, A system of dening relations for the Grigorchuk group , Mat. Zametki 38

(1985), 503–511.

[29] Volodymyr V. Nekrashevych, Cuntz-Pimsner algebras of group actions , 2001, submitted.[30] Peter M. Neumann, Some questions of Edjvet and Pride about innite groups , Illinois J.Math. 30 (1986), 301–316.

[31] Donald S. Passman, Nil ideals in group rings , Michigan Math. J. 9 (1962), 375–384. MR 26#2470

[32] , The algebraic structure of group rings , Wiley-Interscience [John Wiley & Sons], NewYork, 1977, ISBN 0-471-02272-1, Pure and Applied Mathematics.

[33] Victor M. Petrogradskiı, Some type of intermediate growth in Lie algebras , Uspekhi Mat.Nauk 48 (1993), 181–182.

[34] , Intermediate growth in Lie algebras and their enveloping algebras , J. Algebra 179

(1996), 459–482.[35] , Growth of nitely generated polynilpotent Lie algebras and groups, generalized par-

titions, and functions analytic in the unit circle , Internat. J. Algebra Comput. 9 (1999),179–212.

[36] Daniel G. Quillen, On the associated graded ring of a group ring , J. Algebra 10 (1968), 411–418.

[37] Said N. Sidki, On a 2-generated innite 3-group: the presentation problem , J. Algebra 110

(1987), 13–23.[38] , A primitive ring associated to a Burnside 3-group , J. London Math. Soc. (2) 55

(1997), 55–64.[39] John Ronald Reuel Tolkien, The Lord of the Rings , Houghton Mifflin Company, 2002.[40] , The Hobbit , Houghton Mifflin Company, 1999.[41] Ana Cristina Vieira, Modular algebras of Burnside p-groups , Mat. Contemp. 21 (2001), 287–

304, 16th School of Algebra, Part II (Portuguese) (Brasılia, 2000). MR 2004h:20036[42] John S. Wilson, On exponential and uniformly exponential growth for groups , Invent. Math.

155 (2004), 287–303.

Various types of algebraic rings

Documents