Twisted Conjugacy Classes and Residually Finite Groups › workshop › talks › Felshtyn.pdf · Twisted Conjugacy Classes Main resultsFinite groupsAn counter-example R 1 : examples

Main results Finite groups An counter-example R∞: examples and applications Proof of Theorem A Reidemeister spectrum and Theorem A Sketch of a proof of Theorem B Nonabelian cohomology Isogredience classes Rational points Conjectures Twisted conjugacy in connected linear algebraic groups Dynamics

Twisted Conjugacy Classes and Residually FiniteGroups

A. Fel’shtyn

Institute of MathematicsUniversity of Szczecin

http://www.univ.szczecin.pl/felshtyn

Discrete Groups and Geometric Structures , Leuven, Belgium02.06.- 06.06.2014

A. Fel’shtyn University of Szczecin

Twisted Conjugacy Classes


Contents I1 Results, definitions, motivation...2 Finite groups3 An counter-example4 R∞: examples and applications5 Sketch of a proof of Theorem A6 Reidemeister spectrum and Theorem A7 Sketch of a proof of Theorem B8 Nonabelian cohomology

Exact 8-term sequences9 Isogredience classes10 Rational points11 Conjectures12 Twisted conjugacy in connected linear algebraic groups13 Dynamical zeta functions




Main results

I would like to start from the definitions, formulations and examples.

Definition

Reidemeister classes (twisted conjugacy classes, φ-conjugacyclasses) of an automorphism (endomorphism) φ of a (countablediscrete) group G are the classes {g}φ of the equivalence relation

g ∼ xgφ(x−1), g , x ∈ G .

The number of them is the Reidemeister number R(φ).

Example

For φ = Id we have the usual conjugacy classes.




The interest in twisted conjugacy relations has its origins inNielsen-Reidemeister fixed point theory, in Selberg theory, inAlgebraic Geometry, in Galois cohomology , in Representationtheory .




Main results

The following two interrelated problems are among the principalones in the theory of twisted conjugacy (Reidemeister) classes ininfinite discrete groups.The first one is the 20-years-old conjecture on existence of anappropriate twisted Burnside-Frobenius theory (TBFT), i.e.identification of the number R(φ) of Reidemeister classes ofautomorphism φ and the number of fixed points of the inducedhomeomorphism φ on an appropriate dual object (supposingR(φ) <∞).The second one is the 20-years-old problem to outline the class ofR∞ groups (that is R(φ) =∞ for any automorphism φ).




Main results

I will try to explain recent progress in both these problems ( F.-Evgenij Troitsky, arxiv e-print 1204.3175, 2012; KarelDekimpe -Daciberg Goncalves, arxiv e-print 1303.1346, 2013 ; T.Mubeena- P. Sankaran, arxiv e-prints 1111.6181, 2011 and1201.4934, 2012; Timur Nasybullov, arxiv e-print 1201.6515, 2012.)I will try also to explain a essential progress in the third20-years-old problem - the rationality and properties of the Nielsenand the Reidemeister zeta function for infranilmanifolds( KarelDekimpe and Gert-Jan Dugardein, arxiv e-print 1302.5512, 2013; F.- Jong Bum Lee, arxiv e-print 1303.0784, 2013.)




Main results

Definition (Unitary Dual)

Denote by G the set of equivalence classes of unitary irreduciblerepresentations of G and by Gf its part corresponding tofinite-dimensional representations. The class of ρ in G we willdenote by [ρ]. An automorphism φ of G induces a bijectionφ : G → G by the formula [φ(ρ)] := [ρ ◦ φ].

Theorem ( A (F. - Troitsky, 2012))

Let φ : G → G be an automorphism of a residually finite finitelygenerated group G with R(φ) <∞. Then R(φ) is equal to thenumber of φ-fixed points on Gf .




Main results

Definition (Unitary Dual)

Denote by G the set of equivalence classes of unitary irreduciblerepresentations of G and by Gf its part corresponding tofinite-dimensional representations. The class of ρ in G we willdenote by [ρ]. An automorphism φ of G induces a bijectionφ : G → G by the formula [φ(ρ)] := [ρ ◦ φ].

Theorem ( A (F. - Troitsky, 2012))

Let φ : G → G be an automorphism of a residually finite finitelygenerated group G with R(φ) <∞. Then R(φ) is equal to thenumber of φ-fixed points on Gf .




Main results

Theorem ( B (F.- Troitsky, 2012))

Any finitely generated residually finite non-amenable group has theR∞ property (any automorphism has infinitely many twistedconjugacy classes).

This gives a lot of new examples and covers many known classes ofsuch groups.




Conjecture (F. - Troitsky, 2012)

Let φ : G → G be an automorphism of a residually finite finitelygenerated group G with R(φ) <∞. Then G is a solvable - by -finite group.




TBFT, first example

The mentioned conjecture about existence of a twisted and infiniteanalogue of Burnside–Frobenius theorem (TBFT) was formulatedby around 20 years ago in the following way.

Conjecture (F.- Hill, 1993)

Let F (φ) := #Fix(φ). Suppose, R(φ) or F (φ) is finite. Then theycoincide R(φ) = F (φ).

For |G | <∞ and φ = Id it becomes the classicalBurnside-Frobenius theorem.For the simplest infinite group G = Z and its unique non-trivialisomorphism φ = −Id we have

m ∼ k + m − (−k) = m + 2k , ∀ k

Thus even and odd numbers form 2 Reidemeister classes.A. Fel’shtyn University of Szczecin



TBFT, first example

The mentioned conjecture about existence of a twisted and infiniteanalogue of Burnside–Frobenius theorem (TBFT) was formulatedby around 20 years ago in the following way.

Conjecture (F.- Hill, 1993)

Let F (φ) := #Fix(φ). Suppose, R(φ) or F (φ) is finite. Then theycoincide R(φ) = F (φ).

For |G | <∞ and φ = Id it becomes the classicalBurnside-Frobenius theorem.For the simplest infinite group G = Z and its unique non-trivialisomorphism φ = −Id we have

m ∼ k + m − (−k) = m + 2k , ∀ k

Thus even and odd numbers form 2 Reidemeister classes.A. Fel’shtyn University of Szczecin



First example

The dual object can be identified with the unit circle S1 ⊂ C asfollows: each (one dimensional) irreducible representation is definedat m ∈ Z as multiplication by (e iα)m = e iαm. In this way, e iα ∈ S1

corresponds to this representation (denote by ρα). Then

φ(ρα)(m) = (ρα)(−m) = (e iα)−m = (e−iα)m.

Thus, φ coincides with the complex conjugation and has on S1

exactly two fixed points: ±1. Hence, F (φ) = 2 = R(φ).This example shows, in particular, that even for “simple” groups thenumber of twisted conjugacy classes can be finite (in contrast withthe ordinary classes). So, TBFT can be considered as a “correct”generalization of the Burnside-Frobenius theorem to infinite groups.Now we will discuss some other motivations.




First example









First example









Gauss congruences: an application of TBFT

In dynamical context we study Reidemeister numbers of iterations.Suppose, the TBFT holds for a group, and R(φn) <∞. Then,using congruences for fixed points one can obtain the Gausscongruences for Reidemeister numbers. More precisely, let µ be theMobius function:

µ(d) =

1, if d = 1;(−1)k , if d — is a product of

k distinct prime numbers;0, if d is not square free.

Then ∑d |n

µ(d) · R(φn/d ) ≡ 0 mod n, (1)




Gauss congruences: an application of TBFT

Indeed, apply the Mobius’ inversion formula to the following evidentconsequence of Theorem A: R(φn) = # Fix(φn|Gf

) =∑

d |n Pd ,

where Pn denote the number of periodic points of φ on Gf of leastperiod n. We obtain (1) with Pn on the right. But Pn is alwaysdivisible by n, because Pn is exactly n times the number of orbits oflength n.Formula (1) was previously known in the special case of almostpolycyclic groups( F. -Troitsky, Crelle, 2007). The remarkablehistory of the Gauss congruences for integers can be found in thearticle of A. V. Zarelua, On congruences for the traces of powers ofsome matrices. Proceedings of the Steklov Institute ofMathematics, 2008, Vol. 263, p. 78–98. Gauss congruences forLefschetz numbers iterations of a continuous map were proved byDold and Krasnoselski - Zabreiko.




Example: finite groups

Let φ-class functions be functions, which are constant onReidemeister classes of φ, i.e. twisted invariant functions:gf φ(g−1) = f . Evidently, R(φ) is equal to the dimension of thespace of such functions.On the other hand, we have the (two-side equivariant) Peter–Weylisomorphism:

C ∗(G ) ∼=⊕ρ∈G

End Vρ, End Vρ ∼= Mat(dim ρ,C),

(we have written C ∗(G ), but this is C[G ] because the group isfinite).





Let φ-class functions be functions, which are constant onReidemeister classes of φ, i.e. twisted invariant functions:gf φ(g−1) = f . Evidently, R(φ) is equal to the dimension of thespace of such functions.On the other hand, we have the (two-side equivariant) Peter–Weylisomorphism:

C ∗(G ) ∼=⊕ρ∈G

End Vρ, End Vρ ∼= Mat(dim ρ,C),

(we have written C ∗(G ), but this is C[G ] because the group isfinite).





Thus

R(φ) = dim{space of twisted invariant elements of C ∗(G )} =

= +ρ∈G dim{space of twisted invariant elements of End Vρ} =

= +ρ∈G dim{space of intertwinning operators of ρ→ ρ ◦ φ} =

=

{1, if φ[ρ] = [ρ],0, otherwise

= F (φ).





The following nice way of calculation is known (Brauer ?). Considerthe natural action φ∗ on class-functions (for usual conjugacyclasses). Then φ∗ = φ under the identification ofBurnside-Frobenius. The trace of this operator should be the samein the basis of class-functions and in the basis of characters. Inboth cases the operator acts by transpositions of basic elements,thus, its trace is equal to the number of fixed element. HenceF (φ) = the number of φ-invariant usual classes.





The TBFT (in finite case) implies R(φ) = F (φ). Hence, R(φ) =the number of φ-invariant usual classes. The above example with Zshows that this is not correct for infinite groups.With quite another technique the conjecture was proved for Abelianand almost-Abelian (Abelian-by-finite) groups, but unfortunately ...





The TBFT (in finite case) implies R(φ) = F (φ). Hence, R(φ) =the number of φ-invariant usual classes. The above example with Zshows that this is not correct for infinite groups.With quite another technique the conjecture was proved for Abelianand almost-Abelian (Abelian-by-finite) groups, but unfortunately ...




An counter-example

Example (F., E. Troitsky, A. Vershik)

Let G = (Z⊕ Z)×α Z (semi-direct product), where α is defined by

the matrix(

2 11 1

). The automorphism φ is defined on the torus

summand by(

0 −1−1 0

)and on Z as −Id. A complicated but

direct study gives R(φ) = 4. On the other hand, 4finite-dimensional fixed irreducible representations can be find, andat least one infinite-dimensional.

Non-formally speaking, this is caused by bad separateness of thedual object. So, we have two natural ways to overcome:

cutting-off; (of infinite-dimensional part of G )gluing together. (non-separated points of G )




First, Theorem A was proved for almost polycyclic groups ( F. - E.Troitsky, Crelle Journal, 2007). Unfortunately, the following seriesof counter-examples was found (among groups with extremeproperties).Denote by Ff (φ) the number of finite-dimensional fixed points of φ.

Example (Osin’s group)

Osin’s group has exactly 2 usual conjugacy classes. All nontrivialelements of this group G are conjugate. So, the group G is simple,i.e. G has no nontrivial normal subgroup. This implies that group isnot residually finite and so (by Mal’cev’s theorem) it is not linear.Thus, it has no finite-dimensional representations without (group)kernel. Since it is simple, it has no f.d. representations with anon-trivial kernel (except for the trivial 1-dimensional one). Thus,Ff (φ) = 1 for any φ. But R(Id) = 2.

A similar argument remains valid for Ivanov’s group and someA. Fel’shtyn University of Szczecin



HNN-extensions.

R∞: examples

Let us list some classes of R∞-groups known before:non-elementary Gromov hyperbolic groups (A.F.,Levitt-Lustig); relatively hyperbolic groups (A.F.);Baumslag-Solitar groups BS(m, n) except for BS(1, 1)(A.F.–D.Goncalves), generalized Baumslag-Solitar groups, thatis, finitely generated groups which act on a tree with all edgeand vertex stabilizers infinite cyclic (Levitt); the solvablegeneralization Γ of BS(1, n) given by the short exact sequence1→ Z[ 1

n ]→ Γ→ Zk → 1, as well as any groupquasi-isometric to Γ (Taback–Wong);a wide class of saturated weakly branch groups (including theGrigorchuk group and the Gupta-Sidki group) (F. - Yu.Leonov- E.T.), Thompson’s group F (Bleak – F. – Goncalves);generalized Thompson’s groups Fn, 0 and their finite directproducts (Goncalves – Kochloukova);




R∞: examples

symplectic groups Sp(2n,Z), the mapping class groups ModSof a compact oriented surface S with genus g and p boundarycomponents, 3g + p − 4 > 0, and the full braid groups Bn(S)on n > 3 strings of a compact surface S in the cases where Sis either the compact disk D, or the sphere S2 (Damani – F. –Goncalves);extensions of SL(n,Z), PSL(n,Z), GL(n,Z), PGL(n,Z),Sp(2n,Z), PSp(2n,Z), n > 1, by a countable abelian group,and normal subgroup of SL(n,Z), n > 2, not contained in thecentre (Mubeena – Sankaran);GL(n,K ) and SL(n,K ) if n > 2 and K is an infinite integraldomain with trivial group of automorphisms, or K is anintegral domain, which has a zero characteristic and for whichAut(K ) is torsion (Nasybullov);irreducible lattice in a connected semi simple Lie group G withfinite center and real rank at least 2 (Mubeena-Sankaran);




R∞: applications

Evidently, the property R∞ has applications in relation with TBFT.Besides this, one can obtain for example the following.

Theorem (Goncalves–Wong)

For any n ≥ 4 there exists a compact nilmanifold M, dimM = n,such that any homeomorphism f : M → M is homotopic to a fixedpoint free map (for n ≥ 5 even isotopic).




R∞: applications

Evidently, the property R∞ has applications in relation with TBFT.Besides this, one can obtain for example the following.

Theorem (Goncalves–Wong)

For any n ≥ 4 there exists a compact nilmanifold M, dimM = n,such that any homeomorphism f : M → M is homotopic to a fixedpoint free map (for n ≥ 5 even isotopic).




The dual object

Let G be an “appropriate” dual object of G . Take so far G =unitary dual, i.e. the set of equivalence classes of unitary (includinginfinite-dimensional) representations of G . The space G carries thefollowing Jacobson–Fell (or hull-kernel) topology. LetC ∗ρ : C ∗(G )→ B(Hρ) be the representation of C ∗(G ) (= thecompletion of C[G ] w.r.t. the norm of the universal representation)induced by a representation ρ of G . Then the closure of a set Xw.r.t. the hull-kernel topology is

X =

[ρ] : KerC ∗ρ ⊇⋃

[π]∈X

KerC ∗π

.




The dual object

Let G be an “appropriate” dual object of G . Take so far G =unitary dual, i.e. the set of equivalence classes of unitary (includinginfinite-dimensional) representations of G . The space G carries thefollowing Jacobson–Fell (or hull-kernel) topology. LetC ∗ρ : C ∗(G )→ B(Hρ) be the representation of C ∗(G ) (= thecompletion of C[G ] w.r.t. the norm of the universal representation)induced by a representation ρ of G . Then the closure of a set Xw.r.t. the hull-kernel topology is

X =

[ρ] : KerC ∗ρ ⊇⋃

[π]∈X

KerC ∗π

.




The dual object




This topology can be described in terms of weak containment.

Definition

A representation ρ is weakly contained in a representation π (wewrite ρ ≺ π) if diagonal matrix coefficients of ρ can beapproximated by linear combinations of diagonal matrix coefficientsof π uniformly on finite sets. Here a matrix coefficient of arepresentation ρ on a Hilbert space H is the functiong 7→ 〈ρ(g)ξ, η〉 on G for some fixed ξ, η ∈ H, and a diagonal onecorresponds to ξ = η.

Then C ∗Kerπ ⊂ C ∗Ker ρ if and only if ρ ≺ π.




An amenable group may be characterized in several equivalentways, in particular:

There exists an invariant mean on `∞(G ).1G ≺ λG , where 1G is the trivial 1-dimensional representation.C ∗(G ) = C ∗λ(G ).




Sketch of a proof of Theorem A

One can prove that for each finite-dimensional representation ρthere exists a twisted invariant matrix coefficient iff [ρ] is fixed byφ. In this case it is unique (up to scaling) and is defined by

g 7→ Tr(S ◦ ρ(g)), S intertwins ρ and ρ ◦ φ.So, it is sufficient to prove that R(φ) ≤ Ff (φ).




Definition

A group G is residually finite if for any finite set F ⊂ G − {1} thereexists a normal group H of finite index such that H ∩ F = ∅.Taking F formed by g−1

i gj for F0 = {g1, . . . , gs} one obtains anepimorphism G → G/H onto a finite group, which is injective onF0. If a residually finite group G is finitely generated, then H in thedefinition can be supposed to be characteristic.

One more description for R(φ):R(φ) equals the dimension of the space of twisted invariantelements of `∞(G ), i.e. functionals on `1(G ) such that their kernelscontain the closure K1 in `1(G ) of the space of elements of theform b − g [b], g [b](x) := b(gxφ(g−1).Since R(φ) <∞, codimK1 = R(φ), and K1 has a Banach spacecomplement of dimension R(φ). We can take it in a way such thatit has a base ai ∈ C[G ], i = 1, . . . ,R(φ), i.e., all ai ’s have a finitesupport. Let p : G → F = G/H be an epimorphism on a finite




group F such that it distinguishes all elements from the union of(finite) supports of ai and H is characteristic. The image of `1(G )under the induced homomorphism p1 is `1(F ) = C[F ]. Also K1maps epimorphically onto the space Kp of elementsβ − p(g)[β] = p1(b)− p(g)[p1(b)] = p1(b − g [b]) in C[F ]. Thus,{p1(ai )} form a basis of a complement to Kp in C[F ]. Decomposethis (finite dimensional) algebra C[F ] into a direct sum of matrixalgebras, i.e., decompose the left regular representation of F intoirreducible ones: λF ∼= ⊕N

i=1Vi ⊗ V ∗i . Let Ki be formed byx − ρi (g)[x ] in EndVi . Since J is an algebra isomorphism,R(φ) = codimK1 =

∑i codimKi . The last one is 1 if φ(ρi ) = ρi




and 0 otherwise. Thus, R(φ) ≤ Ff (φ).

Reidemeister spectrum

Now we would like to list some cases, where Theorem A isapplicable.Following Romankov, define the Reidemeister spectrum of G asSpec(G ) := {k ∈ N ∪∞ | R(φ) = k for some φ ∈ Aut(G )}. Inparticular, R∞ ⇔ Spec(G ) = {∞}. It is easy to see thatSpec(Z) = {2} ∪ {∞}, and, for n ≥ 2, the spectrum is full, i.e.Spec(Zn) = N ∪ {∞}. For free nilpotent groups we have thefollowing: Spec(N22) = 2N ∪ {∞} (N22 is the discrete Heisenberggroup), Spec(N23) = {2k2|k ∈ N} ∪ {∞} andSpec(N32) = {2n − 1|n ∈ N} ∪ {4n|n ∈ N} ∪ {∞}( Romankov,2009).




Metabelian non-polycyclic groups have quite interestingReidemeister spectrum ( F. - Goncalves, 2011): for example, if θ(1)

is of the form(r 00 s

), then we have the following cases:

a) If r = s = ±1 thenSpec(Z[1/p]2 oθ Z) = {2n|n ∈ N, (n, p) = 1} ∪ {∞} where (n, p)denote the greatest common divisor of n and p.b) If r = −s = ±1 thenSpec(Z[1/p]2 oθ Z) = {2pl (pk ± 1), 4pl |l , k > 0} ∪ {∞}.c) If rs = 1 and |r | 6= 1 thenSpec(Z[1/p]2 oθ Z) = {2(pl ± 1), 4|l > 0} ∪ {∞}.d) If either r or s does not have module equal to one, and rs 6= 1then Spec(Z[1/p]2 oθ Z) = {∞}.




Theorem (Dekimpe-Goncalves, 2013; Romankov, 2011)

Let r > 1 and c ≥ 1 be positive integers.Then Nr ,c has the R∞–property if and only if c ≥ 2r .

An interesting case was studied in F.-Troitsky-Vershik, 2006, wherethe Reidemeister number and the number of fixed points of φwhere compared directly. In this example G is a semidirect productof Z2 and Z by Anosov automorphism defined by the matrix(

2 11 1

). The group G is solvable and of exponential growth. The

automorphism φ is defined by(

0 1−1 0

)on Z2 and as − Id on Z.




Surprise

Theorem (Dekimpe, Goncalves, 2013)

Infinitely generated free group F∞ does not have property R∞ andneither does any of the groups N∞,c , M∞,c or S∞,k .




Sketch of a proof of Theorem B

One has the following number-theoretic inequality. Letx1 ≤ x2 ≤ · · · ≤ xn be positive integers such that∑n

i=1(xi )−1 = 1. Then xn ≤ n2n−1.

Using this, one can estimate the number of fixed points of φ:|CG (φ)| ≤ R(φ)R(φ)−1 for finite groups and even for finitelygenerated residually finite groups.Hence, the stabilizers and relative stabilizers of twisted actionon these groups are finite if R(φ) <∞.

Definition

Denote by γφG the twisted inner representation of G on `2(G ), i.e.

γφG (x)(f )(g) = f (xgφ(x−1)), x , g ∈ G , f ∈ `2(G ).




Sketch of a proof of Theorem B

From the finiteness of stabilizers, one can deduce that γφG isweakly contained in the regular representation λG .With a direct computation we verify that in this case 1G ≺ γφG ,i.e. C ∗Ker γφG ⊂ C ∗Ker 1G .Thus, if G is a finitely generated residually finite group withR(φ) <∞ for some φ, then G is amenable.




Faces of R-infinity

Suppose, a discrete group Γ acts on a discrete group G . We denotethe action σ of s ∈ Γ on g ∈ G by σ(s)(g) := sg . A mapg : Γ→ G , g : s 7→ gs , is called a cocycle, if

gst = gs · sgt .

Two cocycles g and g ′ are cohomological if there exists b ∈ G suchthat

gs = b−1 · g ′s sb. (2)

The corresponding quotient set is H1(Γ,G ) = H1(Γ,G , σ).

Theorem

Reidemeister classes of ϕ are in bijective correspondence withelements of H1(Z,G , σ), in particular, R(ϕ) = #H1(Z,G , σ).




Exact 8-term sequences

Consider a group extension respecting homomorphism φ:

0 // H i //

φ′

��

Gp //

φ

��

G/H //

φ��

0

0 // H i // Gp // G/H // 0,

(3)

where H is a normal subgroup of G .





The following exact sequence (Ph. Heath, 1986) was obtained

1→ Fix(φ′)→ Fix(φ)→ Fix(φ)δ−→ R(φ′)→ R(φ)→ R(φ)→ 1,

(4)where all morphisms are quite evident except of δ, which is definedas follows:

δ(β) = {β−1φ(β)}φ′ , β ∈ G , p(β) = β. (5)





We will give a short proof of this statement using our interpretation.For this purpose consider the diagram

1 // Fix(φ′) // Fix(φ) // Fix(φ)δ // R(φ′) //

OO

��1 // H0(Z,H) // H0(Z,G ) // H0(Z, G )

d // H1(Z,H) //

// R(φ) //OO

��

R(φ) //OO

��

1

// H1(Z,G ) // H1(Z, G ) // 1

(6)

where the bottom exact row can be found e.g. in (Serre, GaloisCohomology). First of all, we should remark that it can beextended to the right by the trivial homomorphism in our case (ofZ-action), because in this case any cocycle is defined by its value at





1 (as it is explained above). Second, the diagram is commutative:due to naturality only the middle square may cause doubts. But themap d is defined as

d(β) = {s 7→ β−1sβ}, where p(β) = β

(see Serre, Galois Cohomology) and it commutes.Our TBFT theorem for an almost polycyclic group G can beinserted in this context as a version of “Poincare–Pontryagin–Tateduality”:

#H1(Z,G , σ) = #H0(Z, Gf , σ), (7)

where Gf is the finite-dimensional part of the unitary dual (a moredelicate result will be discussed in the last section) and one of thesides of this equality is finite. Here σ(1)[ρ] := [ρ ◦ σ]. Aninterpretation of Reidemeister theory in terms of principalhomogeneous spaces (or torsors) seems also prospective.





Isogredience classes

Definition

Suppose, Φ ∈ Out(G ) := Aut(G )/ Inn(G ). We say that α, β ∈ Φare isogredient (or similar) if β = τh ◦ α ◦ τ−1

h for some h ∈ G ,where τh(g) := ghg−1.

Let S(Φ) be the set of isogredience classes of Φ. If Φ = IdG , thenabove α and β are inner, say α = τg , β = τs . Since elements ofcenter Z (G ) give trivial inner automorphisms, we may supposeg , s ∈ G/Z (G ). Then the equivalence relation takes the formτs = τhgh−1 , i.e., s and g are conjugate in G/Z (G ). Thus, S(Id) isthe set of conjugacy classes of G/Z (G ).Denote by S(Φ) the cardinality of S(Φ).





The set of isogredience classes of automorphisms representing agiven outer automorphism and the notion of index Ind(Φ) definedvia the set of isogredience classes are strongly related to importantstructural properties of Φ, for example in another (with respect toBestvina–Handel, 1992) proof of the Scott conjecture( Gaboriau,Levitt, Lustig, 1998).One of the main results of (Levitt, Lustig, 2000) is that for anynon-elementary hyperbolic group and any Φ the set S(Φ) is infinite,i.e., S(Φ) =∞. We will extend this result. First, we introduce anappropriate definition.

Definition

A group G is an S∞-group if for any Φ the set S(Φ) is infinite, i.e.,S(Φ) =∞.

Thus, the above result says: any non-elementary hyperbolic groupis an S∞-group. On the other hand, finite and Abelian groups are





evidently non S∞-groups. Now, let us generalize the abovecalculation for Φ = Id to a general Φ . Two representatives of Φhave form τs ◦ α and τq ◦ α, with some s, q ∈ G . They areisogredient if and only if

τq ◦ α = τg ◦ τs ◦ α ◦ τ−1g = τg ◦ τs ◦ τα(g−1) ◦ α,

τq = τgsα(g−1), q = gsα(g−1)c , c ∈ Z (G ).

So, the following statement is proved.

Lemma

S(Φ) = RG/Z(G)(α), where α is any representative of Φ and α isinduced by α on G/Z (G ).





Theorem

Suppose, |Z (G )| <∞. Then G is an R∞-group if and only if G isan S∞-group.





Rational points

Definition

Let [ρ] ∈ Gf , g 7→ Tg be a (class of a) finite-dimensionalrepresentation. We say that ρ is rational if the number of distinctTg ’s is finite, and irrational otherwise.

Remark

Evidently, ρ is rational if and only if it can be factorized through ahomomorphism G → F on a finite group.





Theorem

Let G be a finitely generated group and for an automorphism φ atleast one of the following two conditions holds:

1). There exist infinitely many finite-dimensional representationclasses in G fixed by φ.

2). There exists an irrational representation ρ fixed by φ.

Then R(φ) =∞.In particular, if we have one of these conditions for everyautomorphism φ, then G has R∞ property.





Conjectures

We would like two finish this section with the following tointerrelated conjectures, motivated by known examples andtheorems.Conjecture R. Let G be a finitely generated residually finite group.Either G is R∞, or G is solvable-by-finite.Conjecture R implies the followingConjecture S. Let G be a finitely generated residually finite groupwith finite center. Either G is S∞, or G is solvable-by-finite.





Twisted conjugacy in connected linear algebraic groups

Let K be an algebraically closed field, over which all algebraicgroups will be taken, K ∗ be the multiplicative group of K and p itscharacteristic exponent (which is the characteristic of K or 1according as the characteristic differs from 0 or not). Let G be aconnected linear algebraic group. This will mean that G is asubgroup of some GLn(K ) and at the same time an algebraic set inthe afflne space determined by the n2 matric coefficients. TheZariski topology, in which the closed sets are the algebraic sets, willbe used. Then G has a unique maximal connected solvable normalsubgroup R , called the radical of G . If G/R is given the structureof an algebraic group, it has a trivial radical, i.e., is semisimple.The semisimple groups turn out to be products, with someamalgamation of (finite) centers, of simple groups, which have beenclassified by Chevalley in the Killing-Cartan tradition .





Lemma (Steinberg(1968))

Let φ : G → G be endomorphism of a connected linear algebraicgroup. The set Fix(φ) is finite if and only if the twisted conjugacyclass of the unit element {e}φ = {xeφ(x−1), x ∈ G} contains adense open part of G .

Theorem (S(Steinberg-1968))

Let φ : G → G be endomorphism of a connected linear algebraicgroup G onto G. If the set Fix(φ) is finite then the twistedconjugacy class of the unit element {e}φ = {xeφ(x−1), x ∈ G} isthe group G i.e. the Reidemeister number R(φ) = 1.





Theorem (Steinberg(1968))

If φ is an automorphism in theorem S and the set Fix(φ) is finite,then connected linear algebraic group G is solvable.





Theorem

Let G be a compact, semisimple, connected linear algebraic group.Then G has the property R∞.

Proof.

Let φ be automorphism of the group G . Then the set Fix(φ) isinfinite by theorem 20. Suppose that the Reidemeister numberR(φ) is finite. For the compact group G , the twisted conjugacyclasses being orbits of twisted action are compact and hence closed.If there is a finite number of them, then they are open as well.Hence the twisted conjugacy classes are open-closed sets thus areconnected components of the connected group G . This implies theReidemeister number R(φ) = 1, thus the twisted conjugacy class ofthe unit element {e}φ is the only one twisted conjugacy class of φ .Then the set Fix(φ) is finite by theorem 18. A contradiction.





Theorem (Steinberg(1968))

Let φ : G → G be endomorphism of a connected linear algebraicgroup G onto G. If R is the radical of G , then φ(R) = R.

Theorem

Let G be a compact, connected linear algebraic group withnontrivial factor group G/R . Then G has the property R∞.

Proof.A. Fel’shtyn University of Szczecin




Consider a group extension respecting homomorphism φ:

0 // R i //

φ′

��

Gp //

φ

��

G/R //

φ��

0

0 // R i // Gp // G/R // 0,

(8)

where, by theorem 22, the radical R is a normal φ-invariantsubgroup of G . Then the induced mapping of Reidemeister classesis an epimorphism, because p(g)p(g)φ(p(g−1)) = p(ggφ(g−1)).By Theorem 21 the nontrivial compact semisimple linear algebraicfactor group G/R has the property R∞. Since the induced mappingof Reidemeister classes is surjective the group G also has theproperty R∞.




Dynamical zeta functions

The Reidemeister zeta functions of f and φ and the Nielsen zetafunction of f are defined as power series:

Rφ(z) = exp

( ∞∑n=1

R(φn)

nzn

),

Rf (z) = exp

( ∞∑n=1

R(f n)

nzn

),

Nf (z) = exp

( ∞∑n=1

N(f n)

nzn

).




Recently, Karel Dekimpe and Gert-Jan Dugardein, arxiv e-print1302.5512, 2013 proved that for any map f on an infra-nilmanifold,the Nielsen number N(f ) of this map is either equal to |L(f )|,where L(f ) is the Lefschetz number of that map, or equal to theexpression |L(f )− L(f+)|, where f+ is a lift of f to a 2-fold coveringof that infra-nilmanifold. By exploiting the exact nature of thisrelationship for all powers of f , they proved that the Nielsen zetafunction for a map on an infra-nilmanifold is always a rationalfunction.




Using this results we are able to write the functional equation forthe Reidemeister zeta function and find a connection between theReidemeister zeta function and the Reidemeister torsion of thecorresponding mapping torus.

Theorem (F.- Jong Bum Lee, 2013)

Let f be a continuous map on an infra-solvmanifold of type R withan affine homotopy lift (d ,D). Assume N(f ) = |L(f )|. Then theNielsen zeta function Nf (z) is a rational function and is equal to

Nf (z) = Lf ((−1)qz)(−1)r

where q is the number of real eigenvalues of D∗ which are < −1and r is the number of real eigenvalues of D∗ of modulus > 1.When the Reidemeister zeta function Rf (z) is defined, we haveRf (z) = Rφ(z) = Nf (z).





Let f be a continuous map on an infra-solvmanifold Π\S of type Rwith an affine homotopy lift (d ,D). Then the Reidemeister zetafunction, whenever it is defined, is a rational function and is equalto

Rf (z) = Nf (z) =

Lf ((−1)nz)(−1)p+nwhen Π = Π+;(

Lf+ ((−1)nz)Lf ((−1)nz)

)(−1)p+n

when Π 6= Π+,





Let f be a continuous map on an orientable infra-solvmanifoldM = Π\S of type R with an affine homotopy lift (d ,D). Then theReidemeister zeta function, whenever it is defined, and the Nielsenzeta function have the following functional equations:

Rf

(1dz

)=

{Rf (z)(−1)mε(−1)p+n

when Π = Π+;Rf (z)(−1)mε−1 when Π 6= Π+

where d is a degree f , m = dimM, ε is a constant in C×,σ = (−1)n, p is the number of real eigenvalues of D∗ which are> 1 and n is the number of real eigenvalues of D∗ which are < −1.If |d | = 1 then ε = ±1.





Let f be a continuous map on an infra-solvmanifold of type R withan affine homotopy lift (d ,D). Then the Nielsen zeta functionNf (z) and the Reidemeister zeta function Rf (z), whenever it isdefined, have the same positive radius of convergence R whichadmits following estimation

R ≥ exp(−h) > 0,

where h = inf{h(g) | g ' f }.If 1 is not in the spectrum of D∗, the radius R of convergence ofRf (z) is

R =1

N∞(f )=

1exp h(f )

=1

(∧

D∗).




Like the Euler characteristic, the Reidemeister torsion isalgebraically defined.Roughly speaking, the Euler characteristic is a graded version of thedimension, extending the dimension from a single vector space to acomplex of vector spaces. In a similar way, the Reidemeister torsionis a graded version of the absolute value of the determinant of anisomorphism of vector spaces.

Theorem (F. - Jong Bum Lee, 2013)

Let f : M → M be a homeomorphism of an infra-nilmanifold M.Assume that N(f ) = |L(f )|. Then

τ(Tf ; p∗E ) =| Lf (λ) |−1=| Nf (σλ) |(−1)r+1=| Rf (σλ) |(−1)r+1

where σ = (−1)p, p is the number of real eigenvalues of F ∗ in theregion (−∞,−1) and r is the number of real eigenvalues of F ∗

whose absolute value is greater that 1.A. Fel’shtyn University of Szczecin




Let f be a homeomorphism on an infra-nilmanifold Π\G with anaffine homotopy lift (d ,D). Then

|Rf ((−1)nλ)(−1)p+n | = |Rφ((−1)nλ)(−1)p+n | = |Nf ((−1)nλ)(−1)p+n |

=

{|Lf (λ)| = τ(Tf ; p∗E )−1 when Π = Π+;|Lf+(λ)Lf (λ)−1| = τ(Tf ; p∗E )τ(Tf+ ; p∗+E )−1 when Π 6= Π+,

where p is the number of real eigenvalues of D∗ which are > 1 andn is the number of real eigenvalues of D∗ which are < −1.





Let f be a homeomorphism on an infra-solvmanifold Π\S of type Rwith an affine homotopy lift (d ,D). Then

|Nf ((−1)nλ)(−1)p+n |

=

{|Lf (λ)| = τ(Tf ; p∗E )−1 when Π = Π+;|Lf+(λ)Lf (λ)−1| = τ(Tf ; p∗E )τ(Tf+ ; p∗+E )−1 when Π 6= Π+,

where p is the number of real eigenvalues of D∗ which are > 1 andn is the number of real eigenvalues of D∗ which are < −1.




Recall the following

Theorem (Bourbaki, Algebra)

Let σ be a Lie algebra automorphism. If none of the eigenvalues ofσ is a root of unity, then the Lie algebra must be nilpotent.

Theorem

If the Reidemeister zeta function Rf (z) is defined for ahomeomorphism f on an infra-solvmanifold M of type R, then M isan infra-nilmanifold.




Gauss congruences

Theorem

Let f be any continuous map on an infra-solvmanifold of type Rsuch that all R(f n) are finite. Then we have∑

d |n

µ(d) R(f n/d ) =∑d |n

µ(d) N(f n/d ) ≡ 0 mod n

for all n > 0.



Twisted Conjugacy Classes and Residually Finite Groups › workshop › talks › Felshtyn.pdf · Twisted Conjugacy Classes Main resultsFinite groupsAn counter-example R 1 : examples

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