aa r X i v : . [ m a t h . G R ] A p r REIDEMEISTER CLASSES IN SOME WEAKLY BRANCH GROUPS
EVGENIJ TROITSKY
Abstract.
We prove that a saturated weakly branch group G has the property R ∞ (anyautomorphism φ : G → G has infinite Reidemeister number) in each of the following cases:1) any element of Out( G ) has finite order;2) for any φ the number of orbits on levels of the tree automorphism t inducing φ isuniformly bounded and G is weakly stabilizer transitive;3) G is finitely generated, prime-branching, and weakly stabilizer transitive with somenon-abelian stabilizers (with no restrictions on automorphisms).Some related facts and generalizations are proved. Introduction
Consider an automorphism φ : G → G of a (countable discrete) group. The Reidemeisternumber R ( φ ) is the number of its Reidemeister or twisted conjugacy classes , i.e. the classesof the following equivalence relation: g ∼ hgφ ( h − ), h, g ∈ G . The Reidemeister class of anelement g we denote by { g } φ .A group has the R ∞ property if R ( φ ) = ∞ for any automorphism φ : G → G . Theproblem of determining of groups having the R ∞ property was raised by A.Fel’shtyn andco-authors in relation with an older conjecture by A.Fel’shtyn and R.Hill [5]: R ( φ ) is equalto the number of fixed points of the associated homeomorphism b φ of the unitary dual b G , ifone of these numbers is finite. This conjecture is called TBFT (twisted Burnside-Frobeniustheorem), because it generalizes to infinite groups and to the twisted case the classicalBurnside-Frobenius theorem: the number of conjugacy classes of a finite group is equal tothe number of equivalence classes of its irreducible representations. The question aboutTBFT formally has a positive answer for R ∞ groups. So, the R ∞ problem is in some sensecomplementary to the TBFT.The TBFT conjecture was proved for finite, abelian and abelian-by-finite groups [5]. Thefurther development, examples, counterexamples and modifications can be found in [11, 9,10, 25, 26, 27].The property R ∞ was proved and disproved for many groups and the number of papers onthe subject and related questions is too large to list all of them and we restrict ourselves togiving reference to several papers and bibliography overview therein: [24, 1, 19, 20, 12, 13,21, 17, 2, 8]. Dynamical aspects of Reidemeister numbers are discussed in [4]. Some directtopological consequences of the property R ∞ for Jiang-type spaces are discussed in [13].In [6] the R ∞ property was proved for a wide class of saturated weakly branch groups.In the present paper we develop these results and prove in Theorem 2.2 that if any auto-morphism of a saturated weakly branch group G is a composition of an inner automorphism Mathematics Subject Classification.
Key words and phrases.
Reidemeister number, R ∞ -group, twisted conjugacy class, residually finite group,weakly branch group.This work is supported by the Russian Science Foundation under grant 16-11-10018. and of a finite order automorphism, then G has the property R ∞ . In particular, this theoremholds for the Grigorchuk group and for the Gupta-Sidki group. In some specific cases theresult can be obtained from [17].We introduce the property WST (Definition 1.7) and prove that if for any automorphism φ of a saturated weakly branch WST group G , induced by an automorphism t of the tree,i.e. φ ( g ) = tgt − , restrictions of t on levels have a uniformly bounded number of orbits, then G has the property R ∞ (Theorem 3.3).In Theorem 4.2 we prove the R ∞ property without any restrictions on the structure ofthe automorphism group of a finitely generated saturated weakly branch WST group G , butwith the restriction on branching numbers to be prime and with an additional restriction onstabilizers.We prove that a saturated weakly branch group on a spherically symmetric tree, such thatany level stabilizer contains an odd permutation at some level, is an R ∞ group (Theorem5.2). Acknowledgement:
The author is indebted to A. Fel’shtyn and the MPIM for help-ful discussions in the Max-Planck Institute for Mathematics (Bonn) in February, 2017, toV. Manuilov for useful suggestions, and to A. Jaikin-Zapirain for a bibliography reference.This work is supported by the Russian Science Foundation under grant 16-11-10018.1.
Preliminaries
First, we recall some necessary facts about Reidemeister classes.
Lemma 1.1.
Any Reidemeister class of φ is formed by some φ -orbits.Proof. Indeed, φ ( g ) = g − gφ ( g ). (cid:3) Definition 1.2.
Denote by τ g the inner automorphism : τ g ( x ) = gxg − .From the equality xyϕ ( x − ) g = x ( yg ) g − ϕ ( x − ) g = x ( yg )( τ g − ◦ ϕ )( x − )we immediately obtain the following statement. Lemma 1.3.
A right shift by g ∈ G maps Reidemeister classes of φ onto Reidemeisterclasses of τ g − ◦ ϕ , In particular, R ( τ g ◦ φ ) = R ( φ ) . Lemma 1.4 ([7, Prop. 3.4]) . Suppose, φ is an automorphism of a finitely generated residuallyfinite group. Let R ( φ ) = r < ∞ . Then the number of fixed elements of φ is bounded by afunction depending only on r . Now we pass to groups acting on trees and give some known and new definitions and facts.Let T be a spherically symmetric rooted tree. This means that all vertexes of the samelevel have the same number of immediate descendants ( branching index ).Denote by D ( v ) the set of immediate descendants of a vertex v ∈ T .A group G acting faithfully on a rooted tree is a weakly branch group , if for any vertex v of T , there exists an element of G which acts nontrivially on the subtree T v with the rootvertex v and trivially outside this subtree. In other words, the rigid stabilizer Rist v of anyvertex v is non-trivial.Evidently a faithful tree group is residually finite. EIDEMEISTER CLASSES IN WEAKLY BRANCH GROUPS 3
We will denote by St( v ) the stabilizer of a vertex v ∈ T ; and by St j the stabilizer of level L j , i.e. St j = ∩ v ∈ L j St( v ).A group G is saturated if, for every positive integer n , there exists a characteristic subgroup H n ⊂ G acting trivially on the n -th level of T and level transitive on any subtree T v with v in the n -th level. Theorem 1.5 ([18]) . Suppose, G is a saturated weakly branch group on a tree T . Then itsautomorphism group Aut G coincides with the normalizer of G in the full group of isome-tries Iso ( T ) of the rooted tree T : every automorphism φ of the group G is induced by theconjugation by an element t from the normalizer and the centralizer of G in Iso ( T ) is trivial. Definition 1.6.
For a group G acting on T and any vertex v ∈ T denote by G { v } thesubgroup of all elements g ∈ G fixing v and all vertexes of T from the next level, except ofimmediate descendants of v .In other words, if v ∈ L j , then G { v } = \ w ∈ L j , w = v \ u ∈ D ( w ) St( u )Thus, Rist v ⊂ G { v } ⊂ St j . Definition 1.7.
We call a group G acting on T weakly stabilizer transitive (WST) if forany vertex v one can find a vertex v ∈ T v such that G { v } acts transitively on immediatedescendants of v . Remark 1.8. If G acts level-transitively, then G { v } are pairwise isomorphic for v from thesame level. Also, they pairwise commute and we can introduce the following well-definedgroup Γ { i } . Definition 1.9.
Denote Γ { i } := Y v ∈ L i − p i ( G { v } ) , where p i : G → G/ St i is the natural projection.Let t be an automorphism of a tree T . Let Orb i ( t ) be the number of orbits of t at thelevel L i . Evidently,1) Orb i ( t ) is a not-decreasing function of i ;2) a fixed vertex of t may be only a successor of a fixed vertex;3) if there is a fixed vertex at the level i + 1, then Orb i +1 ( t ) > Orb i ( t ).So, we have two possibilities:(a) Orb i ( t ) −→ ∞ as i → ∞ ;(b) Orb i ( t ) is bounded. In this case, there is no fixed vertices starting some level, by 3)above.Finally, we will need the following statement from the Galois theory (see, e.g. [3, Sect.3.5]): Lemma 1.10.
A solvable transitive subgroup of the symmetric group S p , where p is prime,is isomorphic either to Z p , or to Z p ⋊ Z p − . In particular, it is either abelian, or containsan odd permutation (a generator of Z p − ). EVGENIJ TROITSKY Finite order automorphisms and around
Lemma 2.1.
Let φ : G → G be an automorphism of oder n < ∞ of a weakly branch groupwith φ ( g ) = tgt − . Then there exists j such that for any j > j there exists an element g j ∈ St j and a number i > j such that { g j } φ ∩ St i = ∅ .Proof. It is sufficient to find an element g j such that hg j th − t − = e at the level L i for any h ∈ G , or equivalently(1) g j t = h − th. By the condition, t has at some level an orbit of length n , and does not have a longerorbit. Take for j the first time when t has on L j an orbit of length n . Then the orbits ofsuccessors also will have the length n . Consider any j > j and an orbit of length n in L j .Let v ∈ L j be a vertex from this orbit. Using the weak branching property we can find anon-trivial element g j ∈ Rist( v ). Let i be the first level where g j acts non trivially, say at v ∈ T v (see Fig. 1). Then b b b b b b b b bb b bb v g j v L i L i − L j t -orbits Figure 1. (2) ( g j t ) n ( v ) = g j t n ( v ) = g j ( v ) = v, because the t -orbit of v has the form v, t ( v ) , . . . , t n − ( v ), t n ( v ) = v and t ( v ) , . . . , t n − ( v )
6∈ T v implying gt ( v ) = t ( v ) , . . . , gt n − ( v ) = t n − ( v ). So, g j t has an orbit of length > n and cannot be conjugate to t at the level L i . We obtain (1). (cid:3) Theorem 2.2.
Suppose, G is a saturated weakly branch group and each automorphism from Out( G ) is of finite order. Then G has the R ∞ property. EIDEMEISTER CLASSES IN WEAKLY BRANCH GROUPS 5
Proof.
By Lemma 1.3 it is sufficient to verify R ( φ ) = ∞ for some φ of finite order n .By Theorem 1.5 φ ( g ) = tgt − for an automorphism t of the tree. Then Lemma 2.1 givesinductively an infinite sequence of representatives of distinct Reidemeister classes. Thus R ( φ ) = ∞ . (cid:3) Example 2.3.
The most studied branch groups – the Grigorchuk group [14] and the Gupta-Sidki group [16] – have outer automorphisms of finite order [15, 23].
Example 2.4.
A more evident example is the group of all isometries of a symmetric rootedtree. In this case all automorphisms are inner.3.
Finite number of orbits
Now we consider the opposite case, when the number of orbits t on L i is uniformly bounded.We will need to restrict ourselves to the WST case. Lemma 3.1.
Let φ : G → G be an automorphism of a WST group with φ ( g ) = tgt − , where t is an automorphism of the tree T . Suppose, t satisfies (b) above, namely, max i Orb i ( t ) = M < ∞ . Then there exists j such that for any j > j there exists an element g j ∈ St j anda number i > j such that { g j } φ ∩ St i = ∅ .Proof. Let j be the level of stabilization of the number of orbits, i.e., Orb j − ( t ) < M andOrb j ( t ) = M , hence Orb j ( t ) = M for any j > j . Note that the lengths of orbits of t at next levels, are the multiples of lengths of orbits on L j (with the coefficient equal tothe appropriate product of branching numbers) and an orbit of smallest length (not uniquegenerally) lies under a smallest orbit on L j .Now take an arbitrary j > j and consider an orbit of t of the smallest size on L j . Let v be a vertex from this orbit, and find by the WST property an element v ∈ T v , v ∈ L i − for some i , with a transitive action of G { v } on its immediate successors. Let v be one ofthese successors. Then, as it was explained, its t -orbit has the smallest length among theorbits on L i . This length is equal to m · b , where m is the length of t -orbit of v and b isthe branching number of v . Choose g j ∈ G { v } such that g j t m ( v ) = v (see Fig. 2). By thedefinition of G { v } , g j t ( v ) = t ( v ) , ( g j t ) ( v ) = t ( v ) , . . . ( g j t ) m ( v ) = g j t m ( v ) = v . Hence, the smallest length of a ( g j t )-orbit on L i is m < m · b =the smallest length of a t -orbiton L i . Thus, g i t and t can not be conjugate and we arrive to (1) and the same argument asin the beginning of the proof of Lemma 2.1, completes the proof. (cid:3) Remark 3.2.
By Lemma 1.1 R ( φ ) < ∞ , if the number of φ -orbits is finite. But the numberof φ -orbits in G is rather weakly related to the number of t -orbits on T . For example, for t = Id, this depends on “how saturated G is”.Similarly to the proof of Theorem 2.2, one can deduce from Lemma 3.1 the followingstatement. Theorem 3.3.
If a saturated weakly branch group G is a WST group and each its non-trivialouter automorphism has the properties from Lemma 3.1, then G is an R ∞ group. Example 3.4.
We do not expect interesting examples of groups here, moreover, we needthe results of this section mostly as a tool for proofs with using for some automorphisms inthe next section (case b) below).Nevertheless, Example 2.4 works here too.
EVGENIJ TROITSKY b b b b b b b b bb b bb vv g j t m ( v ) v L i L i − L j t -orbitspart of t -orbit Figure 2. The general case
Lemma 4.1.
Let φ : G → G be an automorphism of a group G acting level-transitively ona spherically symmetric tree T , with φ ( g ) = tgt − . Suppose,(1) G is finitely generated;(2) G is a WST group;(3) moreover, for an infinite subsequence { i k } of the sequence of levels, arising as transi-tivity levels in the definition of WST, the corresponding group Γ { i } (see Def. 1.9) isnot abelian;(4) branching numbers are prime (may be distinct for distinct levels).Suppose, R ( φ ) < ∞ . Then there exists j such that for any j > j there exists an element g j ∈ St j and a number i > j such that { g j } φ ∩ St i = ∅ .Proof. As in the proof of Lemma 2.1, it is sufficient to find an element g j ∈ St j such that atthe level L i for any h ∈ G (3) g j = h − tht − . Consider two cases:a) Orb i ( t ) → ∞ ;b) Orb i ( t ) is bounded. Case a).
Since R ( φ ) < ∞ and G is finitely generated, by Lemma 1.4 the number of φ -fixed elements for the quotient G/St i is strictly less Orb i − ( t ) at each level i greater some j . EIDEMEISTER CLASSES IN WEAKLY BRANCH GROUPS 7
Now for any j > j , let i − > j be the number of a level with transitive action of G { v } for any v ∈ L i − (see Remark 1.8) such that Γ { i } is not abelian.Each of the above-mentioned φ -fixed elements (except of e ) acts non-trivially at somevertex w s . Thus an element, which fixes these vertexes, is not φ -fixed. Hence there exists v ∈ L i − such that for any v in its t -orbit, p i ( G { v } ) does not contain φ -fixed points, where p i : G → G/St i . Suppose, the t -orbit of v has some length k ( k = 1 can occur in particular).Then p i G { t m ( v ) } = t m p i G { v } t − m , m = 0 , , . . . , k − . Evidently, elements of these groups commute, and we can form a groupΓ := p i ( G { v } ) · · · p i G { t k − ( v ) } with an action of φ . Each γ ∈ Γ acts trivially on all w s . Hence, Γ has no nontrivial φ -fixedelements. So, Γ is a subgroup with a fixed-point-free automorphism φ . Then it is solvableby [22].Hence, its subgroup p i ( G { v } ) is also solvable. It is a transitive subgroup of the symmetricgroup S p , where p is the prime branching number for vertexes from L i − . Then, by Lemma1.10, it is either abelian, or contains an odd permutation p i ( g j ) A p , g j ∈ G { v } . In thefirst case, Γ { i } is abelian in a contradiction with the supposition. In the second case, g j istrivial on L i except the successors of v . So it is an odd permutation on the entire L i , while h − tht − is an even one. This gives (3). Case b).
This case immediately follows from Lemma 3.1. (cid:3)
Similarly to the proof of Theorem 2.2 we obtain from Lemma 4.1 the following statement.
Theorem 4.2.
Suppose, G is a finitely generated saturated weakly branch WST group on aspherically symmetric tree with prime branching numbers and an infinite sequence of non-abelian Γ { i } (i.e. satisfying the suppositions of Lemma 4.1). Then G is an R ∞ group. Remark 4.3.
Reasonable examples will be given in the next section for a version of thisstatement, namely Theorem 5.2.5.
Some generalizations
Evidently the above statements can be easily extended to some more general cases (withmore complicated formulations).For example, Theorem 2.2 can be evidently generalized in the following way.
Theorem 5.1.
Suppose, G is a weakly branch group and each automorphism from Out( G ) is of finite order and defined by an automorphism of the tree. Then G has the R ∞ property. Now we will give another version of Theorem 4.2.
Theorem 5.2.
Suppose, G is a saturated weakly branch group on a spherically symmetrictree, such that for any j , St j contains an element g j defining an odd permutation at somelevel j > j . Then G is an R ∞ group.Proof. Indeed, (3) keeps, because h − tht − is an even permutation and g j is an odd permu-tation at the level j . (cid:3) Example 5.3.
The full isometry group as in Example 2.4 satisfies the conditions of Theorem5.2.
EVGENIJ TROITSKY
Example 5.4.
Consider a saturated weakly branch group G and consider a group Γ gen-erated by G and an infinite series of isometries g j , e.g., transpositions of two neighbouringelements at level L j +1 and somehow defined at their successors. Then Γ satisfies the condi-tions of Theorem 5.2. References [1]
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Dept. of Mech. and Math., Moscow State University, 119991 GSP-1 Moscow, Russia
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