On elementary amenable bounded automata groups
aa r X i v : . [ m a t h . G R ] J a n ON ELEMENTARY AMENABLE BOUNDED AUTOMATAGROUPS
KATE JUSCHENKO, BENJAMIN STEINBERG, AND PHILLIP WESOLEK
Abstract.
There are several natural families of groups acting on rootedtrees for which every member is known to be amenable. It is, how-ever, unclear what the elementary amenable members of these familieslook like. Towards clarifying this situation, we here study elementaryamenable bounded automata groups. We are able to isolate the elemen-tary amenable bounded automata groups in three natural subclasses ofbounded automata groups. In particular, we show that iterated mon-odromy groups of post-critically finite polynomials are either virtuallyabelian or not elementary amenable.
Contents
1. Introduction 12. Preliminaries 53. Reduced form 84. Technical results 115. Groups containing odometers 156. Kneading automata groups 217. Generalized basilica groups 288. Groups of abelian wreath type 359. Examples 41References 451.
Introduction
A group G is amenable if there exists a finitely additive translation in-variant probability measure on all subsets of G . This definition was givenby J. von Neumann, [23], in response to the Banach-Tarski and Hausdorffparadoxes. He singled out the property of a group which forbids paradoxicalactions.The class of elementary amenable groups , denoted by EG , was introducedby M. Day in [6] and is defined to be the smallest class of groups thatcontains the finite groups and the abelian groups and is closed under taking Date : January 9, 2018.The second author was supported by NSA MSP subgroups, quotients, extensions, and directed unions. The fact that theclass of amenable groups is closed under these operations was already knownto von Neumann, [23], who noted at that at that time there was no knownamenable group which did not belong to EG .No substantial progress in understanding the class of elementary amenablegroups was made until the 80s. C. Chou, [5], showed that all elementaryamenable groups have either polynomial or exponential growth. Shortlythereafter, R. Grigorchuk, [9], gave an example of a group with intermediategrowth, and such groups are necessarily amenable but by Chou’s work arenot elementary amenable. Grigorchuk’s group served as a starting point indeveloping the theory of groups with intermediate growth.In the same paper, Chou showed that every infinite finitely generatedsimple group is not elementary amenable. In [14], it was shown that thetopological full group of a Cantor minimal system is amenable, and by theresults of H. Matui, [16], this group has a simple and finitely generatedcommutator subgroup, in particular, it is not elementary amenable. Thiswas the first example of an infinite finitely generated simple amenable group.The strategy of Matui has been extended in [4] and [17] to minimal actions of Z d and groupoids, respectively. This has produced more examples of finitelygenerated simple amenable groups in combination with results of [13], whereit is demonstrated that there are minimal actions of Z on the Cantor setwith amenable topological full groups.Groups of automorphisms of rooted trees were the only source of groupsof intermediate growth until a recent result of V. Nekrashevych, [20], whofound finitely generated, torsion subgroups of the topological full group ofintermediate growth. These groups cannot act on rooted tree, since theyare simple.Summarizing the above, the currently known sources of non-elementaryamenable groups are groups acting on rooted trees and topological fullgroups of Cantor minimal systems.There has been considerable progress in establishing amenability of groupsacting on rooted trees. L. Bartholdi, V. Kaimanovich, and Nekrashevychin [2] proved that groups acting on rooted trees with bounded activity areamenable; these are the so-called bounded automata groups . This was ex-tended to groups of linear activity by G. Amir, O. Angel, and B. Vir´ag in[1], and in [15], Nekrashevych, M. de la Salle, and the first named authorextended this result further to groups of quadratic activity. These resultsproduce large families of amenable groups, but they do not identify thenon-elementary amenable members of the families. The first named au-thor established what seems to be the first results studying the elementaryamenable groups that act on rooted trees in [12]. The present article picksup this thread of research again. We here consider the elementary amenablebounded automata groups. N ELEMENTARY AMENABLE BOUNDED AUTOMATA GROUPS 3
Question 1.1.
What do elementary amenable bounded automata groupslook like? How close are they to abelian?Question 1.1 in full generality seems out of reach at the moment. We an-swer here the question for three large classes of bounded automata groups,which contain many of the hitherto studied examples. Two of these classesare new, and they appear to be interesting in their own right. We addi-tionally present several examples which place restrictions on how one mightspecify “close to abelian.”1.1.
Iterated monodromy groups.
The iterated monodromy groups ofpost-critically finite polynomials, see [18, Chapter 5], form a compelling andnatural class of bounded automata groups. The study of these groups hasgenerated a significant body of research in both dynamics and group theory.See, for example, [8, 19].For this class of groups, we answer the motivating question completely.
Theorem 1.2 (See Corollary 6.11) . Every iterated monodromy group of apost-critically finite polynomial is either virtually abelian or not elementaryamenable.
Generalized basilica groups.
For a finite set X , let X ∗ be the freemonoid on X . By identification with its Cayley graph, X ∗ is naturally arooted tree. For g ∈ Aut( X ∗ ) and v ∈ X ∗ , the section of g at v is denotedby g v ; see Section 2 for the relevant definitions. Definition 1.3.
We say that G ≤ Aut( X ∗ ) is a generalized basilicagroup if G admits a finite generating set Y such that for every g ∈ Y either g x = 1 for all x ∈ X or g x ∈ { , g } for all x ∈ X with exactly one x suchthat g x = g . Remark 1.4.
The classical basilica group can be represented as a general-ized basilica group. The representation is given by considering the naturalaction of the basilica group on [4] ∗ , the 4-regular rooted tree, by restrictingthe action on the binary tree to even levels; see Example 3.11.A key feature of generalized basilica groups is a reduction theorem, The-orem 7.7, which reduces the analysis to self-replicating groups. Using thisreduction, we answer the motivating question for a class of generalized basil-ica groups for which we have some control over the “cycle structure” of thegenerators. Definition 1.5.
Let G = h Y i ≤ Aut( X ∗ ) be a generalized basilica group.We say that G is balanced if for each g ∈ Y such that there is x ∈ X forwhich g x = g , the least n ≥ g n fixes x is also such that g n fixes X .Note that the basilica group acting on [4] ∗ is balanced. Theorem 1.6 (See Corollary 7.18) . Every balanced generalized basilica groupis either (locally finite)-by-(virtually abelian) or not elementary amenable.
KATE JUSCHENKO, BENJAMIN STEINBERG, AND PHILLIP WESOLEK
Examples 9.1 show our theorem for balanced groups cannot be sharpened.While a proof eludes us, we expect the following conjecture to hold.
Conjecture 1.7.
Every generalized basilica group is either (locally finite)-by-(virtually abelian) or not elementary amenable
The class of generalized basilica groups encompasses the torsion-free boundedautomata groups, suggesting it is a natural class.
Proposition 1.8 (See Corollary 3.9) . If G ≤ Aut( X ∗ ) is a torsion-freebounded automata group, then there is a torsion-free generalized basilicagroup H and k ≥ such that H is isomorphic to a subgroup of G and G ֒ → Sym( X k ) ⋉ H X k . Proposition 1.9 (See Corollary 3.10) . Every torsion-free self-replicatingbounded automata group admits a faithful representation as a generalizedbasilica group.
We also exhibit countably many non-isomorphic balanced generalizedbasilica groups; see Examples 9.1. The class of balanced generalized basilicagroups, and hence the class of generalized basilica groups, is thus as large asit can be, since there are only countably many bounded automata groups.1.3.
Groups of abelian wreath type.
Let π : Aut( X ∗ ) → Sym( X )be the homomorphism induced by the action of Aut( X ∗ ) on X . Under thenatural identification with finitary elements, we regard Sym( X ) ≤ Aut( X ∗ ),when convenient. Definition 1.10.
We say that G ≤ Aut( X ∗ ) is of abelian wreath type if π ( G ) ≤ Sym( X ) is abelian and G admits a finite self-similar generating set Y such that for every g ∈ Y either g x = 1 for all x ∈ X or g x ∈ { g } ∪ π ( G )for all x ∈ X with exactly one x such that g x = g . Remark 1.11.
The Gupta-Sidki groups and the more general GGS groupsare of abelian wreath type.A group G ≤ Aut( X ∗ ) of abelian wreath type embeds into the inverselimit ( · · ·≀ X A ≀ X A ) where A is the abelian group π ( G ). This class of boundedautomata groups seems to be among the simplest such groups, because thestructure of these groups is minimally complicated by permutation-group-theoretic phenomena.We answer the motivating question completely for self-replicating groupsof abelian wreath type. Theorem 1.12 (See Theorem 8.13) . Every self-replicating group of abelianwreath type is either virtually abelian or not elementary amenable.
The infinite dihedral group shows that Theorem 8.13 is sharp; see Exam-ple 9.2. The GSS groups are self-replicating as soon as they are infinite, soTheorem 1.12 applies to these groups. Additionally, as there are countablymany self-replicating GGS groups, there are countably many isomorphismtypes of self-replicating groups of abelian wreath type.
N ELEMENTARY AMENABLE BOUNDED AUTOMATA GROUPS 5
Groups containing odometers.
An element d ∈ Aut( X ∗ ) is said tobe an odometer if h d i acts transitively on X and d x ∈ { , d } for all x ∈ X with exactly one x ∈ X such that d x = d . Odometers arise somewhat natu-rally in bounded automata groups. For instance some iterated monodromygroups will contain these; see Example 5.9. On the other hand, our theo-rem for groups with odometers plays an important technical role in otherarguments, so one is welcome to regard this theorem as a useful technicaltool. Theorem 1.13 (See Theorem 5.14) . Every bounded automata group thatcontains an odometer is either virtually abelian or not elementary amenable.
Let G be a bounded automata group containing a level transitive element d . The element d is well-known to be conjugate in the full automorphismgroup of the rooted tree to an odometer, but the conjugator need not pre-serve the property of G being a bounded automata group. In Theorem 1.13,we need the element to be an odometer in the bounded representation of G .2. Preliminaries
For a group G acting on a set X and Y ⊆ X , we write G ( Y ) for the point-wise stabilizer of Y in G . For g, h ∈ G , the commutator [ g, h ] is ghg − h − .2.1. Bounded automata groups.
For a finite set X of size at least 2, let X ∗ be the free monoid on X ; the identity is the empty word. By identifi-cation with its Cayley graph, X ∗ is naturally a rooted tree, but in practiceand in the work at hand, it is often more useful to think of X ∗ as words.We write X k to denote the words of length k . An automorphism g of X ∗ is a bijection such preserves the prefix relation. That is, a is a prefix of b ifand only if g ( a ) is a prefix of g ( b ). Automorphisms of X ∗ act on X ∗ fromthe left, and we denote the group of automorphisms by Aut( X ∗ ).For each k ≥
1, the group Aut( X ∗ ) acts on the set of words of length k .We let π k : Aut( X ∗ ) → Sym( X k ) denote the induced homomorphism. It issometimes convenient to see π k (Aut( X ∗ )) ≤ Aut( X ∗ ), and this is achievedas follows: Say σ ∈ π k (Aut( X ∗ )) and v ∈ X l with l ≥ k . Writing v = xw with x ∈ X k and w ∈ X ∗ , we define σ ( v ) := σ ( x ) w . One verifies that thisidentification gives that π k (Aut( X ∗ )) ≤ Aut( X ∗ ).For a word w ∈ X ∗ , the words with prefix w , wX ∗ , are isomorphic to X ∗ by the map ψ w : X ∗ → wX ∗ defined by ψ w : x wx . For g ∈ Aut( X ∗ ) and w ∈ X ∗ , the element g induces an automorphism g w ∈ Aut( X ∗ ) defined by g w := ψ − g ( w ) ◦ g ◦ ψ w . The automorphism g w is called the section of g at w . A straightforwardcomputation shows the operation of taking sections enjoys two fundamentalproperties. Lemma 2.1.
Let X be a non-empty finite set, g , h , and k be elements of Aut( X ∗ ) , and v and v be elements of X ∗ . Then, KATE JUSCHENKO, BENJAMIN STEINBERG, AND PHILLIP WESOLEK (a) g v v = ( g v ) v (b) ( hk ) v = h k ( v ) k v Remark 2.2.
The previous lemma ensures that map Aut( X ∗ ) × X ∗ → Aut( X ∗ ) by ( g, v ) g v is a cocycle.An automorphism g ∈ Aut( X ∗ ) is called finitary if there is some k ≥ g v = 1 for all v ∈ X k . The least k such that g v = 1 for all v ∈ X k iscalled the depth of the finitary element g . An automorphism g ∈ Aut( X ∗ )is called directed if there is some k ≥ w ∈ X k such that g w = g and g v is finitary for all v ∈ X k \ { w } . The least such k is called the period of g , and the vertex w ∈ X k is called the active vertex of g on X k and isdenoted by v g . We say that g is strongly active if g ( v g ) = v g . We say g is strongly directed if there exists k ≥ w ∈ X k such that g w = g and g u = 1 for all u ∈ X k \ { w } . An element d ∈ Aut( X ∗ ) is said to be an odometer if h d i acts transitively on X and d x ∈ { , d } for all x ∈ X withexactly one x ∈ X such that d x = d .For the work at hand, the following features of automorphisms of rootedtrees play an integral role. Definition 2.3.
An automorphism g ∈ Aut( X ∗ ) is finite state if S ( g ) := { g v | v ∈ X ∗ } is finite. The element g is bounded if there is N ≥ n ≥ |{ v ∈ X v | g v = 1 }| ≤ N. Finitary automorphisms and directed automorphisms are clearly boundedand finite state. A partial converse also holds.
Proposition 2.4 ([2, Proposition 2.7]) . An automorphism g ∈ Aut( X ∗ ) isbounded and finite state if and only if there exists m ≥ such that eachsection g v for v ∈ X m is either finitary or directed. We are now prepared to define the bounded automata groups.
Definition 2.5.
A (set) group G ≤ Aut( X ∗ ) is self-similar if g w ∈ G forall w ∈ X ∗ and g ∈ G . Definition 2.6.
A group G ≤ Aut( X ∗ ) is said to be a bounded automatagroup if G is finitely generated and self-similar and every element g ∈ G isbounded and finite state. Remark 2.7.
Bounded automata groups are named as such, because theyare exactly the groups such that the generators are given by a boundedautomaton; see [18, Chapter 1.5].2.2.
Self-replicating groups.
Letting Aut( X ∗ ) ( w ) be the stabilizer of w in Aut( X ∗ ), the map φ w : Aut( X ∗ ) ( w ) → Aut( X ∗ )by g g w is a homomorphism. We call φ w the section homomorphism at w . For a group G ≤ Aut( X ∗ ), we denote the image φ w ( G ( w ) ) by sec G ( w ),and sec G ( w ) is called the group of sections of G at w . N ELEMENTARY AMENABLE BOUNDED AUTOMATA GROUPS 7
Definition 2.8.
A group G ≤ Aut( X ∗ ) is called self-replicating if G isself-similar, acts transitively on X and sec G ( x ) = G for all x ∈ X . Remark 2.9.
The terms “fractal” and “recurrent” have previously ap-peared in the literature in place of “self-replicating.” We understand that“self-replicating” is the currently accepted term.For a self-replicating group G ≤ Aut( X ∗ ), it follows by induction on thelength of v ∈ X ∗ that sec G ( v ) = G for all v ∈ X ∗ . Self-replicating groupsadditionally act transitively on X k for every k , not just on X . Groupsthat act transitively on X k for all k ≥ level transitive . For g ∈ Aut( X ∗ ) an odometer, h g i is level transitive.In the present work, weak forms of self-similarity and self-replication arisefrequently. Definition 2.10.
We say G ≤ Aut( X ∗ ) is weakly self-similar if sec G ( v ) ≤ G for every v ∈ X ∗ . We say that G weakly self-replicating if sec G ( v ) = G for every v ∈ X ∗ .A weakly self-similar (weakly self-replicating) group need not be self-similar (self-replicating). This holds even in the case that G acts transitivelyon X . See Example 9.4 Remark 2.11.
In the work at hand, we will need to analyze subgroups of abounded automata group which are weakly self-similar, but not self-similar.The expert will note that one can conjugate a weakly self-similar group toobtain a self-similar group. However, if one conjugates the supergroup tomake the subgroup in question self-similar, the conjugate often fails to be abounded automata group.2.3.
Elementary amenable groups.Definition 2.12.
The class of elementary amenable groups , denotedby EG, is the smallest class of groups such that the following hold.(1) EG contains all abelian groups and finite groups.(2) EG is closed under taking subgroups, group extensions, quotients,and direct limits.By work of D. Osin, [21], the class EG admits a simpler description. Letus define recursively classes EG α for α an ordinal as follows. • EG is the class of abelian groups and finite groups. • Given EG α , let EG eα be the class of EG α -by-EG groups and let EG lα bethe groups given by a direct limit of groups in EG α . We define EG α +1 :=EG eα ∪ EG lα . • For λ a limit ordinal, we define EG λ := S β<λ EG β . Theorem 2.13 (Osin, [21, Theorem 2.1]) . EG = S α ∈ ORD EG α . In view of Osin’s theorem, the class of elementary amenable groups admitsa well-behaved ordinal-valued rank.
KATE JUSCHENKO, BENJAMIN STEINBERG, AND PHILLIP WESOLEK
Definition 2.14.
For G ∈ EG, we definerk( G ) := min { α | G ∈ EG α } and call rk( G ) the construction rank of G . Corollary 2.15. If G ∈ EG is finitely generated and neither finite norabelian, then there is L E G such that G/L is finite or abelian and rk( G ) =rk( L ) + 1 . Proposition 2.16.
Let G ∈ EG .(1) If H ≤ G , then rk( H ) ≤ rk( G ) . ([21, Lemma 3.1]) (2) If L E G , then rk( G/L ) ≤ rk( G ) . ([21, Lemma 3.1]) (3) If { } → K → G → Q → { } is a short exact sequence, then rk( G ) ≤ rk( K ) + rk( Q ) . ([21, cf. Lemma 3.2]) (4) rk( G × G ) = rk( G ) . (folklore)3. Reduced form
A given representation of a bounded automata group may not always beoptimal. The reduction developed herein produces a canonical representa-tion of a bounded automata group which often makes arguments simpler.This technique appears to be folklore.Given a tree X ∗ and k ≥
1, the group Aut( X ∗ ) acts faithfully on ( X k ) ∗ ,giving an embedding r k : Aut( X ∗ ) ֒ → Aut(( X k ) ∗ ). The map r k is just therestriction of the action of Aut( X ∗ ) to ( X k ) ∗ ⊂ X ∗ (which is a free monoidon X k and hence an | X | k -regular tree). Lemma 3.1.
Suppose that g ∈ Aut( X ∗ ) and k ≥ .(1) For any w ∈ ( X k ) ∗ and g ∈ G , r k ( g w ) = ( r k ( g )) w .(2) If g is finitary, then r k ( g ) is finitary.(3) If g is directed, then r k ( g ) is directed.(4) If g is strongly directed, then r k ( g ) is strongly directed.Proof. Set O := X k and take w ∈ O ∗ . By definition, the section of g at w in X ∗ is g w = ψ − g ( w ) ◦ g ◦ ψ w where ψ v : X ∗ → vX ∗ by x vx . The map r k is restriction to O ∗ , so we see that r k ( g w ) = ( g w ) ↾ O ∗ = ψ − g ( w ) ↾ O ∗ ◦ g ↾ O ∗ ◦ ψ w ↾ O ∗ = ( r k ( g )) w . Hence, (1) holds.If g is finitary, then we can find some n ≥ g w = 1 for all w ∈ O n . Claim (1) shows that ( r k ( g )) w = 1 for all w ∈ O n , so r k ( g ) isfinitary, verifying (2).If g is (strongly) directed, we can find some n ≥ w ∈ O n either g w = g or g w is finitary (trivial). If g w is finitary (trivial), then claims(1) and (2) ensure that ( r k ( g )) w is finitary (trivial). If g w = g , then (1)implies that r k ( g ) = r k ( g w ) = ( r k ( g )) w . We conclude that for every w ∈ O n either ( r k ( g )) w = r k ( g ) or r k ( g ) w is finitary (trivial). Hence, r k ( g ) is a(strongly) directed element of Aut( O ∗ ), establishing (3) and (4). (cid:3) N ELEMENTARY AMENABLE BOUNDED AUTOMATA GROUPS 9
Corollary 3.2. If G ≤ Aut( X ∗ ) is a bounded automata group and k ≥ ,then r k ( G ) ≤ Aut(( X k ) ∗ ) is a bounded automata group.Proof. This is immediate from Proposition 2.4 and Lemma 3.1. (cid:3)
Recall that for each k ≥
1, there is a homomorphism from Aut( X ∗ ) intoSym( X k ) denoted by π k : Aut( X ∗ ) → Sym( X k ). The functions π k and r k enjoy the following relationship: π ◦ r k = π k . Recall also that we may see π k (Aut( X ∗ )) ≤ Aut( X ∗ ) whenever necessary. Definition 3.3.
For G ≤ Aut( X ∗ ) a bounded automata group, we say that G is in reduced form if G admits a finite generating Y such that for every g ∈ Y either g x = 1 for all x ∈ X or g x ∈ { g } ∪ π ( G ) for all x ∈ X with exactly one x such that g x = g . We say that Y is a distinguishedgenerating set for G . We write G = h Y i ≤ Aut( X ∗ ) to indicate that Y isa distinguished generating set for G .Note that one may always assume that Y is a self-similar set by addingeach section from π ( G ) of a directed element of Y to Y . Neither the reducedform nor the distinguished generating set for a given reduced form are uniquein general. However, all bounded automata groups can essentially be put inreduced form. Establishing this requires a result from the literature. Theorem 3.4 ([18, Theorem 3.9.12]) . For G ≤ Aut( X ∗ ) a bounded au-tomata group, there is a finite self-similar set Z ⊆ G consisting of finitaryand directed elements such that for all g ∈ G there is N for which g v ∈ Z for all v ∈ X k and k ≥ N . The directed elements of the set Z in Theorem 3.4 are all strongly directedif and only if in the automaton with state set Z (and the obvious transitions),each non-trivial state belonging to a cycle cannot reach any state off thatcycle except for the trivial state. Theorem 3.5.
For G ≤ Aut( X ∗ ) a bounded automata group, there are afinite set Z ⊆ G and k ≥ such that the following hold:(1) r k ( G ) ≤ Aut(( X k ) ∗ ) is a bounded automata group.(2) Setting H := r k ( h Z i ) , H ≤ Aut(( X k ) ∗ ) is a bounded automata groupin reduced form with distinguished generating set r k ( Z ) . Moreover,if the directed elements of the set from Theorem 3.4 are stronglydirected, then H is generalized basilica.(3) There is an injective homomorphism G → Sym( X k ) ⋉ H X k .Proof. Let W be a finite generating set for G and let Z ⊂ G be as given byTheorem 3.4; in particular, Z is closed under taking sections. By taking acommon multiple of the periods of directed d ∈ Z , we may find m such thatfor each directed d ∈ Z and x ∈ X m the section d x is either finitary or equal to d . Taking n large enough, we may assume that f x = 1 for all x ∈ X n andfinitary f ∈ Z .Let k be a sufficiently large multiple of m so that k ≥ n and g v ∈ Z forall g ∈ W and v ∈ X k . For every directed d ∈ Z and x ∈ X k , either d x is finitary and in π k ( G ) or d x = d . For every finitary f ∈ Z and x ∈ X k , f x = 1. Set O := X k and let r k : Aut( X ∗ ) → Aut( O ∗ ) be the canonicalinclusion. Corollary 3.2 ensures that r k ( G ) is again a bounded automatagroup, verifying (1). It is easy to verify that h Z i ≤ Aut( X ∗ ) is a boundedautomata group using that Z is closed under taking section, so H = r k ( h Z i )is a bounded automata group by a second application of Corollary 3.2.Let us now argue that H ≤ Aut( O ∗ ) is in reduced form. Set Y := r k ( Z );note that Y is closed under takings sections by Lemma 3.1 as Z is closedunder taking sections. Via Lemma 3.1, Y consists of finitary elements anddirected elements, since Z consists of such elements. Say that r k ( f ) ∈ Y is finitary. Lemma 3.1 ensures that f ∈ Z must also be finitary. For each x ∈ O , f x = 1, so in view of Lemma 3.1, ( r k ( f )) x = 1 for all x ∈ O . Say that r k ( d ) ∈ Y is directed. It follows that d ∈ Z must be directed. For x ∈ O such that d x is finitary, we have that d x ∈ Z . Applying again Lemma 3.1,we deduce that ( r k ( d )) x is finitary and an element of π ( H ). For x ∈ O suchthat d x = d , we likewise deduce that ( r k ( d )) x = r k ( d ). For all x ∈ O , it isthus the case that ( r k ( d )) x ∈ { r k ( d ) } ∪ π ( H ). Moreover, if d is stronglydirected, then so is r k ( d ) by Lemma 3.1. We conclude that H is in reducedform and that if each directed element of Z is strongly directed, then H isgeneralized basilica. Claim (2) thus holds.For claim (3), each g ∈ W is such that g x ∈ Z for any x ∈ X k bychoice of k . From Lemma 3.1, we deduce that for each r k ( g ) ∈ r k ( G )and o ∈ O , ( r k ( g )) o ∈ r k ( h Z i ) = H . We define G → Sym( O ) ⋉ H O by g π ( r k ( g ))( r k ( g ) o ) o ∈ O . One easily verifies that this is a well-definedmonomorphism. (cid:3) Using a second fact from the literature, we obtain a useful corollary.
Proposition 3.6 ([18, Proposition 2.11.3]) . If G is a self-replicating boundedautomata group, then Z as in Theorem 3.4 is a generating set for G . Corollary 3.7. If G ≤ Aut( X ∗ ) is a self-replicating bounded automatagroup, then there is k ≥ such that r k ( G ) ≤ Aut(( X k ) ∗ ) is a self-replicatingbounded automata group in reduced form. In particular, G has a faithfulrepresentation as a self-replicating bounded automata group in reduced form. Remark 3.8.
Theorem 3.5 motivates the form of our definitions of general-ized basilica groups and abelian wreath type groups. Our definitions assumethe group is in reduced form to avoid cumbersome reduction steps.We deduce two further corollaries of Theorem 3.5, which elucidate theconnection between torsion-free bounded automata groups and generalizedbasilica groups. In a torsion-free bounded automata group, every finitary
N ELEMENTARY AMENABLE BOUNDED AUTOMATA GROUPS 11 element is trivial and hence every directed element is strongly directed.Theorem 3.5 thus implies the following consequence.
Corollary 3.9. If G ≤ Aut( X ∗ ) is a torsion-free bounded automata group,then there are a torsion-free generalized basilica group H such that H isisomorphic to a subgroup of G and k ≥ such that G ֒ → Sym( X k ) ⋉ H X k . Corollary 3.10. If G ≤ Aut( X ∗ ) is a self-replicating torsion-free boundedautomata group, then G admits a faithful representation as a generalizedbasilica group. Let us conclude this section by working out the reduced form of the clas-sical basilica group.
Example 3.11.
Let a, b ∈ Aut([2] ∗ ) be defined recursively by a ( iv ) := ( iv i = 0 ib ( v ) i = 1 and b ( iv ) := ( v i = 00 a ( v ) i = 1 . The group G := h a, b i ≤ Aut([2] ∗ ) is called the basilica group ; see [10].There is a convenient notation for the generators, representing them as ele-ments of Sym([2]) ⋉ G , called wreath recursion: the wreath recursion of a is a = (1 , b ), and that of b is b = (01)(1 , a ).We see that a = b and b = a , so G is not in reduced form. Consider r ( G ). Set ˜ a := r ( a ) and ˜ b := r ( b ). In view of Lemma 3.1, ˜ a = r ( a ) = r ( a ) = ˜ a , and ˜ b = ˜ b . One further checks that all other sections of ˜ a and ˜ b on [2] are trivial. In wreath recursion, e a = (23)(1 , , , e a ) and e b =(02)(13)(1 , , , e b ). Hence, r ( G ) is in reduced form, and furthermore, thegroup is a generalized basilica group. In fact, r ( G ) is a balanced generalizedbasilica group. 4. Technical results
This section establishes the key technical result of this work, Lemma 4.2,as well as several general technical observations for later use.4.1.
Elementary amenable self-replicating groups.
Let γ ( x , . . . , x n )be a word in the free group on n generators. For a group G , the verbalsubgroup of G given by γ ( x , . . . , x n ) is γ ( G ) := h γ ( g , . . . , g n ) | g i ∈ G i . Note that verbal subgroups are always characteristic subgroups of G .A group is called max-n if every increasing chain of normal subgroupseventually stabilizes. We will only require that all finitely generated virtuallyabelian groups are max-n, but the class of max-n groups is in fact large andcomplicated.Recall that rk( G ) denotes the construction rank of an elementary amenablegroup G . Lemma 4.1.
Suppose that G ≤ Aut( X ∗ ) is an elementary amenable self-replicating group. Let γ and β be words such that G/β ( γ ( G )) is a max-ngroup. Then there are weakly self-replicating normal subgroups M and N of G with(1) N ≤ M , γ ( G ) ≤ M , and β ( γ ( G )) ≤ N ;(2) rk( M ) = rk( γ ( G )) and rk( N ) = rk( β ( γ ( G ))) ; and(3) β ( M/N ) = 1 .Proof.
Fix x ∈ X and let φ x : G ( x ) → G be the section homomorphism at x . Clearly γ ( G ( x ) ) ≤ γ ( G ) ( x ) ≤ G ( x ) , and as φ x is onto, we have γ ( G ) = φ x ( γ ( G ( x ) )) ≤ φ x ( γ ( G ) ( x ) ) ≤ φ x ( G ( x ) ) = G. Similarly, we have β ( γ ( G )) = φ x ( β ( γ ( G ( x ) ))) ≤ φ x ( β ( γ ( G )) ( x ) ) ≤ φ x ( G ( x ) ) = G. We now define two sequences of normal subgroups of G : Set K := γ ( G ), H := β ( γ ( G )) and K n +1 := φ x (( K n ) ( x ) ), H n +1 := φ x (( H n ) ( x ) ). Trivially, H ≤ K , so H n ≤ K n for all n ≥ K ≤ K and H ≤ H , so K n ≤ K n +1 , H n ≤ H n +1 forall n ≥
0, again by induction. Normality follows by induction because φ x is onto. We claim that β ( K n /H n ) = 1 for all n ≥
0. This is trivial for n = 0. Assume that β ( K n /H n ) = 1. Note that ( K n ) ( x ) ∩ H n = ( H n ) ( x ) andso β (( K n ) ( x ) / ( H n ) ( x ) ) ≤ β ( K n /H n ) = 1. As K n +1 /H n +1 is a quotient of( K n ) ( x ) / ( H n ) ( x ) , we deduce that β ( K n +1 /H n +1 ) = 1.Since G/β ( γ ( G )) is a max-n group, there is L such that K L = K L +1 and H L = H L +1 . We claim M := K L and N := H L for such an L satisfy thelemma. The previous paragraph ensures that M and N are normal in G .We see that sec M ( x ) = K L +1 = K L = M , and since the section subgroupsare conjugate to each other, sec M ( y ) = M for all y ∈ X . The obviousinduction argument now gives that sec M ( v ) = M for all v ∈ X ∗ , so M isweakly self-replicating. The same argument shows that N is weakly self-replicating. Claims (1) and (3) follow from the previous paragraph. In viewof Proposition 2.16, induction shows rk( K j ) = rk( K ) = rk( γ ( G )) for all j . The group M is thus such that rk( M ) = rk( γ ( G )). A similar argumentshows rk( N ) = rk( β ( γ ( G ))), completing the proof of (2). (cid:3) By considering the special case that β is a letter, we obtain the following. Lemma 4.2.
Suppose that G ≤ Aut( X ∗ ) is an elementary amenable self-replicating group. Let γ be a word such that G/γ ( G ) is a max-n group.Then there is a weakly self-replicating M E G such that γ ( G ) ≤ M and rk( M ) = rk( γ ( G )) . There are a couple of important corollaries of Lemma 4.2. We say a word w ( x , . . . , x n ) is d -locally finite if every d -generated group G such that w ( G ) = { } is finite. N ELEMENTARY AMENABLE BOUNDED AUTOMATA GROUPS 13
Lemma 4.3 (cf. [7, p. 517]) . For every d -generated finite group A , there isa d -locally finite word w such that w ( A ) = 1 . Corollary 4.4.
Suppose that G ≤ Aut( X ∗ ) is a finitely generated elemen-tary amenable self-replicating group. If G is non-abelian, then there is aweakly self-replicating M E G such that G/M is either finite or abelianand rk( M ) + 1 = rk( G ) . Furthermore, if rk( G ) is witnessed (in the senseof Corollary 2.15) by a normal subgroup N , then G/M is finite if
G/N isfinite and
G/M is abelian if
G/N is abelian.Proof.
That G is self-replicating ensures that G is not finite. Since G isnon-abelian, Corollary 2.15 supplies N E G such that G/N is either finiteor abelian and rk( N ) + 1 = rk( G ).If G/N is abelian, Lemma 4.2 applied to the verbal subgroup [
G, G ] sup-plies the desired subgroup M . Let us suppose that N is of finite index in G . Say that G is d -generated and let γ be a d -locally finite word such that γ ( A ) = 1 where A := G/N , whose existence is given by Lemma 4.3. Wethus have that γ ( G ) ≤ N . It is also that case that G/γ ( G ) is finite, since G/γ ( G ) is a d -generated group and γ ( G/γ ( G )) = 1, so rk( γ ( G ))+1 = rk( G ).Applying Lemma 4.2 to γ ( G ) supplies the desired subgroup M . (cid:3) In the case of a self-replicating elementary amenable group for which therank is given by a finite index subgroup, we can say a bit more.
Corollary 4.5.
Suppose that G ≤ Aut( X ∗ ) is a finitely generated elemen-tary amenable self-replicating group. If G is not virtually abelian and admits M E G with finite index such that rk( M ) + 1 = rk( G ) , then there are weaklyself-replicating subgroups A E G and F E G such that(1) A ≤ F , F/A is abelian, and
G/F is finite; and(2) rk( F ) + 1 = rk( A ) + 2 = rk( G ) .Proof. Since M is finitely generated and neither finite nor abelian, as G isnot virtually abelian, Corollary 2.15 supplies a normal subgroup N E M such that M/N is either finite or abelian and rk( N ) + 1 = rk( M ). It cannotbe the case that M/N is finite, since this contradicts the rank of G , so M/N is abelian. Say that G is d -generated and let γ be a d -locally finite word suchthat γ ( A ) = 1 where A := G/M , whose existence is given by Lemma 4.3.It follows that γ ( G ) ≤ M and G/γ ( G ) is finite. Since M/N is abelian, itfollows that [ γ ( G ) , γ ( G )] ≤ N . Thusrk( G ) ≤ rk( γ ( G )) + 1 ≤ rk( M ) + 1 = rk( G ) , and rk( γ ( G )) ≤ rk([ γ ( G ) , γ ( G )]) + 1 ≤ rk( N ) + 1 = rk( M ) = rk( γ ( G )) . We conclude that rk( γ ( G ))+1 = rk( G ) and rk([ γ ( G ) , γ ( G )])+2 = rk( γ ( G ))+1 = rk( G ).The group G/ [ γ ( G ) , γ ( G )] is a finitely generated virtually abelian (as G/γ ( G ) is finite) and hence a max-n group. Applying Lemma 4.1 with β = [ x, y ], we can find weakly self-replicating F and A normal in G such that A ≤ F , F/A is abelian, γ ( G ) ≤ F (and so G/F is finite), rk( F ) = rk( γ ( G ))and rk( A ) = rk([ γ ( G ) , γ ( G )]). This completes the proof. (cid:3) Generalities on self-similar groups.
The results here are primarilyfor the reader’s convenience, as these results are easy consequences of thedefinitions. We will appeal to these results throughout this work and of-ten without explicit reference. The reader already comfortable with groupsacting on rooted trees can safely skip this subsection.
Lemma 4.6.
Suppose that G ≤ Aut( X ∗ ) , G acts level transitively on X ∗ ,and H E G is weakly self-similar. If H fixes x ∈ X k for some k ≥ , then H is trivial.Proof. That H is normal ensures that H fixes X k . That H contains sec H ( v )for all v ∈ X ∗ implies that H in fact fixes ( X k ) n for all n ≥
1. Hence, H acts trivially on X ∗ . (cid:3) Lemma 4.7.
Suppose that G ≤ Aut( X ∗ ) acts transitively on X and supposethat there is a weakly self-similar M E G such that G/M is abelian or M acts transitively on X . If g ∈ G fixes y and x in X and g x = 1 , then g y ∈ M .Proof. Take f ∈ G such that f ( x ) = y . If M acts transitively on X , we take f ∈ M . The commutator [ f, g − ] is an element of M , and [ f, g − ] y ∈ M since [ f, g − ] fixes y and M is weakly self-replicating. Noting that ( f − ) y =( f x ) − , we see[ f, g − ] y = ( f g − f − g ) y = f x · · ( f x ) − · g y = g y . Hence, g y ∈ M . (cid:3) Lemma 4.8.
Suppose that G ≤ Aut( X ∗ ) acts transitively on X and isweakly self-replicating. If H E G is weakly self-similar and of finite index in G , then H acts transitively on X .Proof. Fix x ∈ X and let φ x : G ( x ) → G be the section homomorphism,which is surjective because G is weakly self-replicating. Since H is weaklyself-similar, the surjective homomorphism ˜ φ x : G ( x ) H/H → G/H by gH φ x ( g ) H is well-defined. The group G ( x ) H/H is thus a subgroup of
G/H thatsurjects onto
G/H . As
G/H is finite, we deduce that
G/H = G ( x ) H/H .Hence, H acts transitively on X . (cid:3) Example: the basilica group.
We close our technical discussion byexhibiting how these results are applied to prove the basilica group is non-elementary amenable. The basic strategy applied here is used throughoutthis work.Let G := h e a, e b i ≤ Aut([4] ∗ ) be the basilica group in the reduced formas given in Example 3.11. The group G is self-replicating; one can verifythis directly or see [10]. The elements e a and e b in wreath recursion are e a = (23)(1 , , , e a ) and e b = (02)(13)(1 , , , e b ). N ELEMENTARY AMENABLE BOUNDED AUTOMATA GROUPS 15
Let us suppose toward a contradiction that G is elementary amenable.One easily verifies that G is not abelian. Corollary 4.4 thus produces aweakly self-replicating subgroup M E G such that rk( M ) + 1 = rk( G ) and G/M is either finite or abelian. We see that e a = (1 , , e a, e a ) and e b =(1 , e b, , e b ). Lemmas 4.7 and 4.8 imply that e a ∈ M and e b ∈ M , so M = G .This contradicts the rank of G . We conclude that G is not elementaryamenable. Remark 4.9.
The general strategy to show a given self-replicating boundedautomata group is not elementary amenable is to contradict the rank of thegroup via the weakly self-replicating subgroups provided by either Corol-lary 4.4 or Corollary 4.5. In the case of the basilica group, the squares e a and e b act trivially on [4] and have some trivial sections, so we may applyLemmas 4.7 and 4.8 to deduce that M = G , which contradicts the rank. Ingeneral, we must work much harder.5. Groups containing odometers
It is convenient to begin with our theorems on bounded automata groupscontaining odometers.
Definition 5.1.
An element d ∈ Aut( X ∗ ) is said to be an odometer if h d i acts transitively on X and d x ∈ { , d } for all x ∈ X with exactly one x ∈ X such that d x = d .In the course of proving these results, we upgrade Corollary 4.5, and theresulting more powerful statement, Corollary 5.11, will be used frequentlyin later sections.5.1. Self-replicating groups.
This preliminary section shows that boundedautomata groups containing an odometer are close to being self-replicating.We begin by characterizing self-replication for generalized basilica groups.
Lemma 5.2.
Suppose that h ∈ Aut( X ∗ ) is such that h x ∈ { , h } for all x ∈ X with at most one x such that h x = h . For any w, v ∈ X and j ∈ Z for which h j ( v ) = w , there is i such that h i ( v ) = w and ( h i ) v = 1 and ≤ | i | < | X | .Proof. If no x ∈ X is such that h x = h , then the lemma is immediate.Suppose there is a unique x ∈ X with h x = h . Choose j with | j | ≥ h j ( v ) = w . If ( h j ) v = 1, we are done. Otherwise, it is the casethat ( h j ) v = h ± , since we chose | j | least. As that cases are similar, let ussuppose that ( h j ) v = h (i.e., j > O be the orbit of x under h h i . The word x is the unique element of X for which h has a non-trivial section, so ( h k ) z = 1 for all z ∈ X \ O and k ∈ Z . We deduce that v ∈ O . Take m ≥ h m fixes x andobserve that ( h m ) y = h for all y ∈ O . The element h − m h j = h j − m is thussuch that h j − m ( v ) = w and ( h j − m ) v = h − h = 1 and | j − m | < | X | . (cid:3) Lemma 5.3.
Suppose that G = h Y i ≤ Aut( X ∗ ) is a generalized basilicagroup and let O be an orbit of G on X . For all x ∈ O , it is then the casethat sec G ( x ) = h Y ′ i where Y ′ is the collection of directed g ∈ Y such that v g ∈ O .Proof. Take d ∈ Y and let W , . . . , W l list the orbits of h d i on O , for some1 ≤ k ≤ n . For w, v ∈ W j , there is some power i such that d i ( v ) = w and( d i ) v = 1 by Lemma 5.2. Plainly, d i G ( v ) d − i = G ( w ) , and we deduce thatsec G ( w ) = ( d i ) v sec G ( v )(( d i ) v ) − . Hence, sec G ( w ) = sec G ( v ), since ( d i ) v = 1. We conclude that sec G ( w ) =sec G ( v ) for all w, v ∈ W j and 1 ≤ j ≤ l . Since G = h Y i and acts transitivelyon O , it follows that sec G ( v ) = sec G ( w ) for all v, w ∈ O .For x ∈ O and h ∈ Y , the section h x is an element of h Y ′ i . Inductingon the word length, one sees that g x ∈ h Y ′ i for all g ∈ G , so a fortiorisec G ( x ) ≤ h Y ′ i . On the other hand, for d ∈ Y ′ , take n ≥ d n fixes v d . The section of d n at v d is d , so d ∈ sec G ( v d ) = sec G ( x ). Weconclude that h Y ′ i = sec G ( x ) for all x ∈ O , verifying the lemma. (cid:3) Proposition 5.4.
For G = h Y i ≤ Aut( X ∗ ) a generalized basilica group, G is self-replicating if and only if G is generated by the directed elements of Y and G acts transitively on X .Proof. Suppose first that G is self-replicating. By definition, G acts transi-tively on X . Fix x ∈ X and let Z ⊆ Y be the directed elements. An easyinduction argument shows that g x ∈ h Z i for all g ∈ G . We deduce that G = sec G ( x ) ≤ h Z i ≤ G , and thus, G is generated by Z .For the converse, let Z be the directed elements of Y . Since G actstransitively on X , sec G ( x ) = h Z i for all x ∈ X by Lemma 5.3. As h Z i = G ,we deduce that sec G ( x ) = G . (cid:3) A version of Proposition 5.4 holds for bounded automata groups withodometers.
Proposition 5.5.
Suppose that G = h Y i ≤ Aut( X ∗ ) is a bounded automatagroup in reduced form. If G is self-replicating, then G is generated by Z := { d x | d ∈ Y is directed and x ∈ X } . The converse holds if G contains an odometer.Proof. Suppose first that G is self-replicating. Fix x ∈ X . An easy inductionargument shows that g x ∈ h Z i for all g ∈ G . We deduce that G = sec G ( x ) ≤h Z i ≤ G , and thus, G is generated by Z .For the converse, let h ∈ G be an odometer. Immediately, we see that G acts transitively on X . For each x and y ∈ X distinct, Lemma 5.2 supplies i such that h i ( x ) = y and ( h i ) x = 1. Fixing x ∈ X , for any g ∈ G and y ∈ X , there are i and j such that h j gh i fixes x and ( h j ah i ) x = g y . Theimage of the section homomorphism sec G ( x ) = φ x ( G ( x ) ) therefore contains N ELEMENTARY AMENABLE BOUNDED AUTOMATA GROUPS 17 all sections g y for g ∈ G and y ∈ X . In particular, sec G ( x ) contains Z , sosec G ( x ) = G . We conclude that G is self-replicating. (cid:3) Our final preliminary lemma shows that one can easily reduce to the groupgenerated by the set Z . Lemma 5.6.
Suppose that G = h Y i ≤ Aut( X ∗ ) is a bounded automatagroup in reduced form. Letting Z := { d x | d ∈ Y is directed and x ∈ X } ,the group h Z i is a bounded automata group in reduced form, and G ֒ → Sym( X ) ⋉ h Z i X . Proof.
That h Z i is a bounded automata group in reduced form is immediate.For each x ∈ X and h ∈ Y , it follows the section h x is an element of Z . Inducting on the word length, one sees that g x ∈ h Z i for all g ∈ G and x ∈ X . The map G → Sym( X ) ⋉ h Z i X given by g π ( g )(( g x ) x ∈ X ) is thus well-defined. One verifies that this mapis also a monomorphism. (cid:3) Generalized basilica groups.Lemma 5.7.
Suppose that G = h Y i ≤ Aut( X ∗ ) is a self-replicating gener-alized basilica group that is non-abelian. If H E G is weakly self-replicatingand G/H is abelian, then
G/H is finite.Proof.
In view of Proposition 5.4, we may assume that Y consists of directedelements.Suppose first that k ∈ Y does not act transitively on X and let n ≥ k n fixes X . There is some x ∈ X such that ( k n ) x = 1, andthere is y ∈ X such that ( k n ) y = k i for some i ≥
1. Applying Lemma 4.7,we conclude that ( k n ) y = k i ∈ H . Hence, k is torsion in G/H .We next consider h ∈ Y that acts transitively on X . Suppose toward acontradiction that h i / ∈ H for all i ≥
1. Fix k ∈ Y \ { h − } and fix y ∈ X such that k ( y ) = y and k y = 1. By Lemma 5.2, there is | X | > | i | ≥ h i k ( y ) = y and ( h i ) k ( y ) = 1. The element h i k has non-trivial sectionsexactly at some x , . . . , x l in X where l ∈ {| i | , | i | + 1 } , since h i has | i | manynon-trivial sections (all equal to h or h − ) and k has one non-trivial section.Say that x is the element of X at which the section of h i k is of the form ak for a ∈ { , h ± } .Suppose there is some x j such that x j and x lie in different orbits of thecyclic group h h i k i acting on X . Let m ≥ h i k ) m fixes x j .The section (( h i k ) m ) x j is of the form h r for some 1 ≤ | r | ≤ i . The element( h i k ) m also fixes y and has a trivial section at y . Applying Lemma 4.7, wededuce that h r ∈ H , which is absurd. It is thus the case that all x j lie inthe same orbit of h h i k i acting on X . Take m ≥ h i k ) m fixes x and observe that (( h i k ) m ) x = h i k . Applying again Lemma 4.7, weconclude that h i k ∈ H . For any other y ′ ∈ X such that k ( y ′ ) = y ′ and k y ′ = 1, the same argumentgives | X | > | j | ≥ h j k ( y ′ ) = y ′ and h j k ∈ H . Thus, h j − i = h j kk − h − i ∈ H . In view of our reductio assumption, we conclude that i = j . Therefore, h i k fixes all y ∈ X such that k ( y ) = y and k y = 1.Say that k fixes x and k x = 1. Letting l ≥ k l fixes v k , Lemma 4.7 implies that k ∈ H . Since h i k is also in H , h i ∈ H , which isabsurd. We conclude that k fixes no x such that k x = 1. The element h i k must then fix all y ∈ X \ { v k } , so h i k fixes X . The subgroup H containsall sections of h i k , since it is weakly self-similar, so if i > i = 1 and v h = k ( v k ), then h ∈ H , which we assume to be false. The element hk therefore fixes X , and ( hk ) v k = hk . It follows that hk = 1. This concludesthe reductio argument, since k = h − .We have now established that every element k ∈ Y is such that k i ∈ H for some i , hence G/H is finite. (cid:3)
Theorem 5.8.
Suppose that G = h Y i ≤ Aut( X ∗ ) is a generalized basilicagroup. If G contains an odometer, then either G is virtually abelian or G isnot elementary amenable.Proof. Without loss of generality, we may assume that Y contains π ( G ) ∩ G .Let K be the subgroup generated by the directed members of Y . The sub-group K equals h Z i where Z is as in Lemma 5.6, so in view of Lemma 5.6, K is virtually abelian if and only if G is virtually abelian. By replacing G with K if needed, we assume that each element of Y is directed. Proposition 5.4now ensures that G is self-replicating.Let us suppose toward a contradiction that G is elementary amenable butnot virtually abelian. Applying Corollary 4.4, there is H E G such that H is weakly self-replicating, rk( H ) + 1 = rk( G ), and G/H is either finite orabelian. In the case that
G/H is abelian, Lemma 5.7 ensures that
G/H isin fact finite. The group G thus admits a weakly self-replicating H E G such that rk( H ) + 1 = rk( G ) and G/H is finite. Since H E G is of finiteindex, Corollary 4.5 supplies a weakly self-replicating A E G such that G/A is virtually abelian and rk( A ) + 2 = rk( G ).Fix h ∈ G such that h is an odometer and form L := A h h i . For each x and y ∈ X distinct, Lemma 5.2 supplies i such that h i ( x ) = y and ( h i ) x = 1.Fixing x ∈ X , for any a ∈ A and y ∈ X , there are i and j such that h j ah i fixes x and ( h j ah i ) x = a y . The image of the section homomorphismsec L ( x ) = φ x ( L ( x ) ) therefore contains all sections of A .For any k ∈ Y , there is some a ∈ A such that a ( k ( v k )) = k ( v k ), byLemma 4.6. The element k − ak is a member of A , and ( k − ak ) v k = a k ( v k ) k .The group sec L ( x ) thus contains a k ( v k ) k and a k ( v k ) . Hence, k ∈ sec L ( x ), andwe deduce that sec L ( x ) = G . On the other hand, A ( x ) E L ( x ) , and L ( x ) /A ( x ) is abelian. Thus, A = sec A ( x ) E sec L ( x ) = G , so G/A is abelian. Thisimplies that rk( G ) = rk( A ) + 1 which is absurd. (cid:3) N ELEMENTARY AMENABLE BOUNDED AUTOMATA GROUPS 19
Example 5.9.
Let G := h e a, e b i ≤ Aut([4] ∗ ) be the basilica group in the re-duced form as given in Example 3.11. The elements e a and e b in wreath recur-sion have the following form: e a = (23)(1 , , , e a ) and e b = (02)(13)(1 , , , e b ).We now see that e a e b − = (23)(1 , , , e a )(02)(13)(1 , e b − , ,
1) = (0312)(1 , e a e b − , , . Hence, e a e b − is an odometer, so G contains an odometer. Theorem 5.8 nowgives a second proof that G is not elementary amenable.5.3. Technical results revisited.Lemma 5.10.
Suppose that G = h Y i ≤ Aut( X ∗ ) is a self-replicatingbounded automata group in reduced form. If G is elementary amenable butnot virtually abelian, then there is a weakly self-replicating M E G such that G/M is finite and rk( M ) + 1 = rk( G ) Proof.
In view of Lemma 3.1, if G is in reduced form, then r k ( G ) ≤ Aut(( X k ) ∗ )is again in reduced form for any k ≥
1. By passing to r ( G ), we may thusassume that | X | > N E G such that G/N is either finite or abelian and rk( N ) + 1 = rk( G ). Suppose that G/N isabelian. We will argue that
G/N is finite. Letting F be the collection offinitary elements of G , we will in fact show that F N is of finite index in G .This suffices to prove the theorem as F is locally finite, and hence G/N willhave a locally finite subgroup of finite index, and hence is finite.Fix d ∈ Y directed and suppose first that h d i does not act transitivelyon X . Let x ∈ X be such that d x = d and y ∈ X be such that y is not inthe orbit of x under h d i . Taking m ≥ d m fixes X , itfollows that ( d m ) x = ( f d ) i for some i ≥ f finitary and ( d m ) y = r for r some finitary. Since G is self-replicating, there is h ∈ G such that h ( y ) = x and h y = 1. The commutator [ h, d − m ] is an element of N , and[ h, d − m ] x = r − ( f d ) i , which is an element of N as N is weakly self-replicating. Since G/N isabelian, it follows that d i ∈ F N .Now suppose that d ∈ Y acts transitively on X . In view of Theorem 5.8,there must be a non-trivial finitary element f ∈ Y . We may find a power n := | X | > i ≥ d i f fixes some x ∈ X . This presents two cases:( d i f ) x is finitary or ( d i f ) x = rdr ′ for r and r ′ finitary. The cases are similar,so we only consider the latter. Recall that n >
2. If i = n −
1, then d − f fixes x , so we replace d with d − . We may thus assume that n − > i .Replace d i with d j where j := i − n and note that 1 < | j | . The sections of d j f have three possible forms: finitary, rdr ′ d − r ′′ with r, r ′ , and r ′′ finitary,or rd − r ′ with r and r ′ finitary. The section ( d j f ) x is of the form rdr ′ d − r ′′ ,and since i < n −
1, there is some y ∈ X such that ( d j f ) y = sd − s ′ with s and s ′ finitary elements. Let k be least such that ( d j f ) k fixes y . Since G is self-replicating, thereis h ∈ G such that h ( x ) = y and h x = 1. The commutator [( d j f ) k , h ] is anelement of N , and [( d j f ) k , h ] y = a k . . . a (( d j f ) x ) − k , where each a i has one of aforementioned forms, ( d j f ) x has the form rdr ′ d − r ′′ and a = (( d j f ) k ) y is of the form sd − s ′ .Since G/N is abelian, the a i of the form rdr ′ d − r ′′ are equal to a fini-tary modulo N . We may thus write [( d j f ) k , h ] y mod N = b . . . b m mod N where each b i is either finitary or of the form sd − s ′ . Commuting the fini-taries to the left modulo N , we see that [( d j f ) k , h ] y mod N = f ′ d − l mod N for f a finitary and l non-zero. On the other hand, [( d j f ) k , h ] y ∈ N since N is weakly self-replicating. It now follows that d l ∈ F N .We have now demonstrated that every z ∈ Y admits l such that z l ∈ F N .Hence,
F N is of finite index in G as G/N is finitely generated abelian. (cid:3)
In view of Corollary 4.5, Lemma 5.10 yields the following consequence.
Corollary 5.11.
Suppose that G ≤ Aut( X ∗ ) is a self-replicating boundedautomata group. If G is elementary amenable but not virtually abelian, thenthere are weakly self-replicating subgroups N E G and M E G such that(1) N ≤ M , M/N is abelian, and
G/M is finite; and(2) rk( M ) + 1 = rk( N ) + 2 = rk( G ) . Corollary 5.11 ensures that various examples of elementary amenablegroups do not have faithful representations as self-replicating bounded au-tomata groups.
Proposition 5.12.
Suppose that G is an elementary amenable group that isnot virtually abelian. If every finite index subgroup has the same rank as G ,then G has no faithful representation as a self-replicating bounded automatagroup.Proof. Suppose for contradiction G has a faithful representation as a self-replicating bounded automata group. Appealing to Corollary 3.7, we mayassume that G ≤ Aut( X ∗ ) is a representation of G as a bounded automatagroup in reduced form. Lemma 5.10 now implies that G has a finite indexsubgroup of lower rank. This contradicts our hypotheses. (cid:3) Corollary 5.13.
The groups A ≀ Z where A is a non-trivial abelian group donot have faithful representations as self-replicating bounded automata groups. Corollary 5.13 is rather interesting because the classical lamplighter group Z / Z ≀ Z does have representations as a self-replicating group [18, cf. Propo-sition 1.9.1] (see also [22] where Z / Z is replaced by an arbitrary finiteabelian group), and on the other hand, Example 9.3 below shows it hasrepresentations as a bounded automata group. However, one can never finda representation which is simultaneously as a self-replicating group and asa bounded automata group N ELEMENTARY AMENABLE BOUNDED AUTOMATA GROUPS 21
The general case.Theorem 5.14.
Suppose that G ≤ Aut( X ∗ ) is a bounded automata group.If G contains an odometer, then either G is virtually abelian or G is notelementary amenable.Proof. Suppose toward a contradiction that G is elementary amenable butnot virtually abelian. Via Theorem 3.5, it is enough to consider G = h Y i ≤ Aut( X ∗ ) to be in reduced form. Without loss of generality, we may assumethat Y contains an odometer h .Setting Z := { d x | d ∈ Y is directed and x ∈ X } , Lemma 5.6 im-plies that H := h Z i is a bounded automata group in reduced form, and G ֒ → Sym( X ) ⋉ H X . Moreover, h ∈ Z , so H contains an odometer andis elementary amenable but not virtually abelian. Proposition 5.5 furtherimplies that H is self-replicating.Applying Corollary 5.11, there is a weakly self-replicating M E H suchthat H/M is virtually abelian and rk( M ) + 2 = rk( H ). Using the odometer h ∈ H , set L := M h h i . For each u and w ∈ X distinct, Lemma 5.2 supplies i such that h i ( u ) = w and ( h i ) u = 1. Fixing v ∈ X , for any m ∈ M and w ∈ X , there are i and j such that h j mh i fixes v and ( h j mh i ) v = m w . Theimage of the section homomorphism sec L ( v ) = φ v ( L ( v ) ) therefore contains m w for every m ∈ M and w ∈ X .For any directed k ∈ Y and x ∈ X the active vertex of k , there is some a ∈ M such that ak ( x ) = k ( x ), by Lemma 4.6. The section ( k − ) ak ( x ) isfinitary of depth one, so( k − ) ak ( x ) x ′ = (( k − ) ak ( x ) ) x ′ = 1for any x ′ ∈ X . The element k − ak is a member of M , and for any y ∈ X ,( k − ak ) xy = ( k − ) ak ( x ) a k ( x ) ( k ( y )) a k ( x ) k ( y ) k y = a k ( x ) k ( y ) k y The group sec L ( v ) thus contains a k ( x ) k ( y ) k y and a k ( x ) k ( y ) . Hence, k y ∈ sec L ( v ). We deduce that Z ⊆ sec L ( v ), so sec L ( v ) = H . On the other hand, M ( v ) E L ( v ) , and L ( v ) /M ( v ) is abelian. Since M = sec M ( v ) E sec L ( v ) = H ,we deduce that H/M is abelian. Therefore, rk( H ) = rk( M ) + 1 which isabsurd. (cid:3) Kneading automata groups
Preliminaries.
For a permutation σ ∈ Sym( X ), a complete cycledecomposition of σ is a cycle decomposition including all length one cycles. Definition 6.1.
Let X be a finite set and α = ( σ , . . . , σ n ) be a sequenceof permutations of X . Say that each σ i has complete cycle decomposition c i, . . . c i,k i . We define the cycle graph of α , denoted by Γ α , as follows: V Γ α := X ⊔ { ( i, j ) | ≤ i ≤ n and 1 ≤ j ≤ k i } and E Γ α := {{ ( i, j ) , x } | x appears in c i,j } . The graph Γ is a bipartite graph with bipartition consisting of X and C := { ( i, j ) | ≤ i ≤ n and 1 ≤ j ≤ k i } . Some authors define the cyclegraph using only non-trivial cycles. We allow trivial cycles to make ourdiscussion more streamlined later. Allowing for trivial cycles only addsleaves to the graph. In particular, these leaves to do affect whether or notthe cycle graph is a tree. Definition 6.2.
For X a finite set, a finite sequence α of elements of Sym( X )is called tree-like if the cycle graph is a tree.We will find tree-like sequences in the definition of kneading automatagroups. Let us note several facts for later use. Lemma 6.3. If ( σ , . . . , σ n ) is a tree-like sequence of permutations of afinite set X , then H := h σ , . . . , σ n i acts transitively on X .Proof. First note that if x, x ′ ∈ X are connected by a path of length 2, thenthey belong to a cycle of some σ i and are hence in the same orbit of H .Since the cycle graph Γ is bipartite, and hence any path between elementsof X decomposes as a composition of length two paths of the above sort, H is transitive by the connectivity of Γ. (cid:3) For a permutation σ ∈ Sym( X ), we denote the collection of fixed pointsin X of σ by fix( σ ). Lemma 6.4 ([11, Lemma 6.5]) . Let X be a finite set and α = ( σ , . . . , σ n ) be a tree-like sequence of permutations of X . Then(1) For any i = j , | fix( σ i ) | + | fix( σ j ) | ≥ .(2) If | fix( σ i ) | + | fix( σ j ) | ≤ for some i = j , then σ k = 1 for all k / ∈{ i, j } . Kneading automata.
A set Y ⊆ Aut( X ∗ ) is called self-similar if g x ∈ Y for every x ∈ X and g ∈ Y . Definition 6.5.
We say that G ≤ Aut( X ∗ ) is a kneading automatagroup if G admits a finite self-similar generating set Y such that the fol-lowing hold:(1) For each non-trivial h ∈ Y , there is a unique g ∈ Y and x ∈ X suchthat g x = h .(2) For each h ∈ Y and each cycle ( x . . . x n ) of π ( h ), possibly of lengthone, there is at most one 1 ≤ i ≤ n such that h x i = 1.(3) The sequence ( π ( h )) h ∈ Y \{ } is tree-like.We call Y a distinguished generating set and write G = h Y i ≤ Aut( X ∗ )to indicate that Y is a distinguished generating set for G .A straightforward verification shows that a kneading automata group isa bounded automata group. Furthermore, kneading automata groups natu-rally arise in the study of iterated monodromy groups. N ELEMENTARY AMENABLE BOUNDED AUTOMATA GROUPS 23
Theorem 6.6 (Nekrashevych, [18, Theorem 6.10.8]) . Every iterated mon-odromy group of a post-critically finite polynomial has a faithful representa-tion as a kneading automata group
Kneading automata groups enjoy several useful properties.
Lemma 6.7. If G = h Y i ≤ Aut( X ∗ ) is a kneading automata group, then G ≤ Aut( X ∗ ) is a self-replicating bounded automata groupProof. To see that G is self-replicating, we must verify that G acts tran-sitively on X and the section homomorphisms are onto for every x ∈ X .Since ( π ( y )) g ∈ Y \{ } is tree-like, π ( G ) acts transitively on X , so G acts on X transitively.To see that the section homomorphisms are onto, we first argue that foreach v, w ∈ X , there is k ∈ G such that k ( v ) = w with k v = 1. Fix g ∈ Y and say that { O , . . . , O n } lists the orbits of h g i on X . Take v, w ∈ O i andfix | j | ≥ g j ( v ) = w . If ( g j ) v = 1, we are done, so let ussuppose that ( g j ) v is non-trivial. By the definition of a kneading automatagroup, there is a most one z ∈ O i such that g z is non-trivial. Say that g z = h and let m ≥ g m fixes z . The element g m fixes O i and( g m ) u = h for every u ∈ O i . On the other hand, ( g j ) v = h ± , since we chose j with | j | least. Either g j − m or g j + m , depending on the sign of ( g j ) v , thensends v to w with trivial section at v .For an arbitrary v, w ∈ X , there is a word γ = g i n n . . . g i with g k = g k +1 ∈ Y such that γ ( v ) = w . We may further assume that Σ nj =1 | i j | is leastamong all such words γ . The image g i ( v ) lies in the orbit of v under h g i .Appealing to the previous paragraph, we may replace g i , if necessary, with g i ± m i for some m i such that that g i ± m i ( v ) = g i ( v ) and ( g i ± m i ) v = 1.Doing this for each g j as necessary, we produce a new word γ ′ such that γ ′ ( v ) = w and ( γ ′ ) v = 1.Fix x ∈ X and let g ∈ Y . By the definition of a kneading automatagroup, there is h ∈ Y and z ∈ X such that h z = g . The previous paragraphsupplies γ and δ such that γ ( x ) = z with γ x = 1 and δ ( h ( z )) = x with δ h ( z ) = 1. The element δhγ fixes x , and( δhγ ) x = δ h ( z ) h z γ x = h z = g. The image φ x ( G ( x ) ) of the section homomorphism at x thus contains Y . Weconclude that φ x ( G ( x ) ) = G , and G is self-replicating. (cid:3) In addition to self-replication, kneading automata groups enjoy a robust-ness property. Specifically, the representation r k ( G ) is again a kneadingautomata group, for any k ≥
1. This was proved in [18, Proposition 6.7.5]using the dual automaton and a topological argument; here we provide asimple counting argument.
Lemma 6.8 ([18, Proposition 6.7.5]) . If G = h Y i ≤ Aut( X ∗ ) is a kneadingautomata group, then (1) for all k ≥ , r k ( G ) = h r k ( Y ) i ≤ Aut(( X k ) ∗ ) is a kneading automatagroup;(2) there is k ≥ such that r k ( G ) = h r k ( Y ) i ≤ Aut(( X k ) ∗ ) is a kneadingautomata group in reduced form.In particular, every kneading automata group has a faithful representationas a kneading automata group in reduced form.Proof. For any k , conditions (1) and (2) of the definition of a kneadingautomata group are easily verified to hold for r k ( G ). To prove (3), we showthat ( π k ( y )) y ∈ Y \{ } is a tree-like sequence in Sym( X k ) for any k ≥
1. We usethat a finite connected graph is a tree if and only if it has Euler characteristic1. Set Z := Y \ { } Let us begin by making several observations. Let Γ i be the cycle graphof ( π i ( y )) y ∈ Z . Define l i : Z → N by setting l i ( g ) to be the number of orbitsof h g i on X i . For g ∈ Z , let Y g,i := { g x | x ∈ X i } \ { } . As G is a kneadingautomata group, it follows that Z = ⊔ g ∈ Z Y g,i .We observe that the functions l i +1 are defined in terms of l and l i . Fix g ∈ Z . The difference l i ( g ) − | Y g,i | is the number of orbits O of h g i on X i such that g x = 1 for every x ∈ O . For each such orbit O and x ∈ X , theset of words Ox is an orbit of h g i on X i +1 . There are thus | X | ( l i ( g ) − | Y g,i | )many orbits of h g i on X i +1 of this type. For any other orbit O , let h ∈ Y g,i be the non-trivial section of g for some x ∈ O ; the orbit O for a given h isunique. For any W an orbit of h h i acting on X , the argument in the secondparagraph of the proof of Lemma 6.7 (which only uses conditions (1) and(2) of a kneading automata group) shows that OW is an orbit of h g i actingon X i +1 . We conclude that l i +1 ( g ) = | X | ( l i ( g ) − | Y g,i | ) + X h ∈ Y g,i l ( h ) . Note further that Γ i is connected for every i , since G acts transitively on X i . Additionally, | V Γ i | = | X | i + X g ∈ Z l i ( g ) and | E Γ i | = | Z || X | i . We now argue by strong induction on i ≥ π i ( y )) y ∈ Z isa tree-like sequence. The base case holds by definition. Suppose ( π i ( y )) y ∈ Z is a tree-like sequence for all i ≤ k . To verify the inductive claim, it sufficesto show that | V Γ k +1 | − | E Γ k +1 | = 1.The inductive hypothesis ensures that X g ∈ Z l i ( g ) = | Z || X | i − | X | i + 1for any i ≤ k . For k + 1, we have | V Γ k +1 | = | X | k +1 + X g ∈ Z l k +1 ( g ) . N ELEMENTARY AMENABLE BOUNDED AUTOMATA GROUPS 25
In view of the relationship between the functions l i , we substitute to obtain | V Γ k +1 | = | X | k +1 + P g ∈ Z (cid:16) | X | ( l k ( g ) − | Y g,k | ) + P h ∈ Y g,k l ( h ) (cid:17) = | X | k +1 + | X | P g ∈ Z l k ( g ) − | X || Z | + P g ∈ Z l ( g )= | X | k +1 + | X | k +1 | Z | − | X | k +1 + | X | − | X || Z | + | Z || X | − | X | + 1= | X | k +1 | Z | + 1 . Hence, | V Γ k +1 | − | E Γ k +1 | = 1, as required. Therefore, r k ( G ) is kneadingautomata group.Lemma 6.7 ensures that G is a self-replicating bounded automata group,and by Corollary 3.7, there is k such that r k ( G ) is in reduced form. Hence, r k ( G ) is a kneading automata group in reduced form. (cid:3) Elementary amenable kneading automata groups.
We begin byextracting a result from the proof of [11, Theorem 8.2]. For g ∈ Aut( X ∗ ),let fix k ( g ) be the collection of fixed points of h g i acting on X k . Lemma 6.9.
Let G = h Y i ≤ Aut( X ∗ ) be a kneading automata group inreduced form with | X | > . If there are distinct g and h in Y such that | fix ( g ) | + | fix ( h ) | = 2 , then either(1) | fix ( g ′ ) | + | fix ( h ′ ) | > for all distinct, non-trivial g ′ and h ′ in Y ,or(2) G is the infinite dihedral group.Proof. Since G is self-replicating, it is infinite. Suppose that (1) fails. Saythat g ′ and h ′ in Y are such that | fix ( g ′ ) | + | fix ( h ′ ) | ≤
3. Lemma 6.8 tellsus that r ( G ) is a kneading automata group, so Lemma 6.4 implies that π ( d ) acts trivially on X for all d ∈ Y \ { g ′ , h ′ } . If h or g is an element of Y \ { g ′ , h ′ } , then h or g fixes X , which contradicts that | fix ( g ) | + | fix ( h ) | =2. We conclude that { g, h } = { g ′ , h ′ } . In particular, | fix ( g ) | ≤ | fix ( h ) | ≤ G is in reduced form, all finitary elements of Y have depth one, sothere are no non-trivial finitary elements in Y \ { g, h } , since all elements of Y \ { g, h } act trivially on X . If d ∈ Y \ { g, h } is directed, then the previousparagraph ensures that it fixes X . Every section d x for x ∈ X \ { v d } istrivial where v d is the active vertex for d on level one, as d fixes X . Theelement d therefore acts trivially on X ∗ , and d = 1. Hence, Y = { g, h, } .Take Γ the cycle graph for ( π ( g ) , π ( h )) and set n := | X | . Let K g and K h be the number of non-trivial cycles in g and h , respectively, and F g and F h be the number of trivial cycles in g and h , respective. Euler’s formulafor trees implies that | V Γ | − | E Γ | = 1. In view of the definition of Γ, we seethat | V Γ | = n + K g + F g + K h + F h and | E Γ | = 2 n . Therefore, K g + K h = n −
1. Observe that | fix ( g ) ∪ fix ( h ) | =2, as G = h g, h i acts transitively on X and hence g, h do not have a commonfixed point. If fix ( g ) ∪ fix ( h ) = { x, x ′ } , then it follows that every element of X \{ x, x ′ } belongs to both a non-trivial cycle of π ( g ) and a non-trivial cycle of π ( h ) and x, x ′ each belong to a non-trivial cycle of exactly one of π ( g )or π ( h ). It follows that 2 n − ≥ K g + 2 K h with equality if and only if π ( g ) , π ( h ) are both products of disjoint two-cycles in their respective cycledecompositions (omitting trivial cycles). As K g + K h = n −
1, we deduce that π ( g ) and π ( h ) are products of disjoint two-cycles, so | π ( g ) | = | π ( h ) | = 2.We now have two cases: (a) Y contains a non-trivial finitary element, and(b) Y contains no non-trivial finitary element. For case (a), suppose that g isfinitary. The element g therefore has order two, since G is in reduced form.If g fixes a point u ∈ X , then g must fix pointwise uX in X , since g u = 1.This implies that | fix ( g ) | >
3, which is absurd. We conclude that g actsfixed point freely on X , so h fixes two points v , v of X . If some h v i = 1,then h fixes v i X contradicting that | fix ( h ) | ≤
3. Since G is a kneadingautomata group with self-similar generating set Y = { g, h, } , we deducethat, up to relabelling, h v = h , h v = g and all other sections of h aretrivial. Since π ( h ) is a product of disjoint two-cycles and g is an involution,it now follows that h = 1. Hence, G is an infinite dihedral group.For case (b), both g and h are directed, and as Y is self-similar, neither g nor h has non-trivial finitary sections. Take u ∈ X \ { v g } . If g fixes u , then | fix ( g ) | >
3, since g u = 1; this is absurd. The only possible fixed point of g in X is its active vertex v g . Likewise, the only possible fixed point of h in X is its active vertex v h . Therefore, g fixes v g , and h fixes v h . We concludethat g and h are torsion with | g | = | h | = 2. The group G is thus infiniteand generated by two involutions, so G is an infinite dihedral group. (cid:3) We are now ready to prove the desired theorem.
Theorem 6.10.
Suppose that G = h Y i ≤ Aut( X ∗ ) is a kneading automatagroup. If G is elementary amenable, then G is virtually abelian.Proof. Suppose toward a contradiction that G is not virtually abelian. Webegin with several reductions. In view of Lemma 6.7, G is a self-replicatingbounded automata group, and by Lemma 6.8, we may assume that G is inreduced form. By passing to r ( G ), where r : Aut( X ∗ ) → Aut(( X ) ∗ ) is thecanonical inclusion, we may assume that | X | >
3. If g, h ∈ Y are distinctelements such that | fix ( g ) | + | fix ( h ) | = 2, then Lemma 6.9 ensures that | fix ( g ′ ) | + | fix ( h ′ ) | > g ′ and h ′ in Y . Passing to r ( G ) a second time, we may additionally assume that | fix ( g ) | + | fix ( h ) | > g and h in Y .Applying Corollary 5.11, we obtain weakly self-replicating M E G and F E G such that rk( M ) + 2 = rk( G ) = rk( F ) + 1, M ≤ F , F/M is abelian,and
G/F is finite. By Lemma 4.8, F also acts transitively on X . We nowhave two cases (1) every g ∈ Y fixes at least two points on X , and (2) someunique g ∈ Y fixes one or fewer points on X .Let us suppose first that every g ∈ Y has at least two fixed points. Fix h ∈ Y directed. Since h fixes at least two points, we may find x ∈ X suchthat h fixes x and h x =: σ is finitary. The section σ is again an elementof Y , so it fixes some x ′ ∈ X , as it also fixes two points. The element h N ELEMENTARY AMENABLE BOUNDED AUTOMATA GROUPS 27 therefore fixes xx ′ , and h xx ′ = σ x ′ = 1, since G is in reduced form. Let v h be the active vertex of h in X and d ≥ h d fixes v h . For O the orbit of v h under h h i , that G is a kneading automata group ensuresthat v h is the only element of O for which h has a non-trivial section. Itfollows that ( h d ) v h = h . Since ( h d ) also fixes xx ′ , Lemma 4.7 implies that h ∈ F .For any non-trivial section h z of h , the definition of the kneading automataensures that h z is the only non-trivial section of h on the orbit for z under h h i . Taking k ≥ h k fixes z , it follows that ( h k ) z = h z .Recalling that F is weakly self-similar, we conclude that h z ∈ F . It nowfollows from Proposition 5.5 (as G is self-replicating) that F = G , whichcontradicts the rank of G .Suppose next that there is a unique g ∈ Y that has at most one fixedpoint on X . Fix h ∈ Y \ { g } directed. The element h has at least threefixed points in X , so there is x ∈ X such that h fixes x , h x is finitary, and h x = g . The section h x fixes some x ′ ∈ X , as it differs from g . Hence, h fixes xx ′ , and h xx ′ = 1. Just as in the previous case, it now follows that h ∈ F .The quotient F/M is abelian, and F acts on X transitively. As Lemma 4.7also applies in this setting, we can run the argument again to deduce that h ∈ M , and as in the previous case, h x ∈ M for all x ∈ X . We concludethat h x ∈ M for all x ∈ X and h ∈ Y \ { g } . The subgroup M thereforecontains Y \ ( { g x | x ∈ X } ∪ { g } ).Let us now consider the element g . The quotient G/M cannot be abelianbecause rk( G ) = rk( M ) + 2 and so { g x | x ∈ X } ∪ { g } is not equal to { , g } . The element g thereby admits non-trivial finitary sections and, inparticular, is not, itself, finitary. Fix x ∈ X such that g x is a non-trivialfinitary element σ and note that σ ∈ Y \ { g } fixes a point x ′ ∈ X .Let d ≥ g d fixes v := v g and let m ≥ g m fixes x . Since G is a kneading automaton group, and henceeach orbit of g has at most one non-trivial section, ( g d ) v = g and ( g m ) x = σ . Observe that g d fixes vv , g m fixes xx ′ ( g m ) xx ′ = 1 and ( g d ) vv = g .The element g d m thus fixes vv and xx ′ . Moreover, ( g d m ) xx ′ = 1 and( g d m ) vv = g m . Applying Lemma 4.7, we deduce that g m ∈ F . The element g d m is therefore also in F , and since F/M is abelian, a second application ofLemma 4.7 ensures that g m ∈ M . The element g m fixes x , and ( g m ) x = σ .Therefore, σ ∈ M , since M is weakly self-similar. We conclude that M contains every finitary section of g and hence Y \ { g } . The quotient G/M is thus abelian, which contradicts the rank of G . (cid:3) Theorems 6.6 and 6.10 yield an immediate corollary.
Corollary 6.11.
Every iterated monodromy group of a post-critically finitepolynomial is either virtually abelian or not elementary amenable. Generalized basilica groups
Definition 7.1.
We say that G ≤ Aut( X ∗ ) is a generalized basilicagroup if G admits a finite generating set Y such that for every g ∈ Y either g x = 1 for all x ∈ X or g x ∈ { , g } for all x ∈ X with exactly one x such that g x = g . We call Y a distinguished generating set and write G = h Y i ≤ Aut( X ∗ ) to indicate that Y is a distinguished generating set for G .A straightforward verification shows that a generalized basilica group isa bounded automata group.7.1. A reduction theorem.
The reduction result established herein re-duces many questions for generalized basilica groups to the self-replicatingcase. It follows by induction on the number of directed elements of a distin-guished generating set.Our first lemma gives a tool to reduce to groups with possibly fewernumber of directed generators, based on the orbits of the group on the firstlevel of the tree.
Lemma 7.2.
Suppose that G = h Y i ≤ Aut( X ∗ ) is a generalized basilicagroup, let O , . . . , O n list the orbits of G on X , and let Y i ⊆ Y be thecollection of g ∈ Y such that v g ∈ O i . For each i , h Y i i ≤ Aut( X ∗ ) is ageneralized basilica group generated by directed elements, and G ֒ → n Y i =1 Sym( O i ) ⋉ n Y i =1 h Y i i O i . Proof.
That each h Y i i ≤ Aut( X ∗ ) is a generalized basilica group generatedby directed elements is immediate.For each x ∈ O i and h ∈ Y , the section h x is an element of h Y i i . Inductingon the word length, one sees that g x ∈ h Y i i for all g ∈ G and x ∈ O i .Letting π : G → Sym( X ) be the induced homomorphism, we see that π ( G ) ≤ Q n Sym( O i ). We may now define G → n Y i =1 Sym( O i ) ⋉ n Y i =1 h Y i i O i . by g π ( g )(( g x ) x ∈ O i ) ni =1 . One verifies that this function is a monomor-phism. (cid:3) Our next lemmas address the case in which Lemma 7.2 does not reducethe number of directed elements.
Lemma 7.3.
Suppose that G = h Y i ≤ Aut( X ∗ ) is a generalized basilicagroup. If O ⊆ X is setwise preserved by the action of G on X , then O ∗ issetwise preserved by G , and the induced homomorphism φ : G → Aut( O ∗ ) issuch that φ ( G ) is a generalized basilica group with distinguished generatingset φ ( Y ) . N ELEMENTARY AMENABLE BOUNDED AUTOMATA GROUPS 29
Proof.
That G is self-similar ensures that O ∗ is setwise preserved by G . Take g ∈ G and w ∈ O ∗ . By definition, g w = ψ − g ( w ) ◦ g ◦ ψ w where ψ v : X ∗ → vX ∗ by x vx . The map φ is nothing but restriction to O ∗ , so we see that φ ( g w ) = ( g w ) ↾ O ∗ = ψ − g ( w ) ↾ O ∗ ◦ g ↾ O ∗ ◦ ψ w ↾ O ∗ = φ ( g ) w . It now follows that φ ( G ) ≤ Aut( O ∗ ) is a generalized basilica group withgenerating set φ ( Y ). (cid:3) We call the action of G on O ∗ a sub-basilica action of G induced from X ∗ . Lemma 7.4.
Let O ⊆ X and let G be the subgroup of Aut( X ∗ ) consistingof those elements that fix O ∗ pointwise and have trivial sections outside of O ∗ . Then G ∼ = Sym( X \ O ) O ∗ and hence is locally finite.Proof. It is clear that G is a subgroup of Aut( X ∗ ) since if f, g ∈ G and w / ∈ O ∗ , we have ( f g ) w = f g ( w ) g w = 1 as g ( w ) / ∈ O ∗ . There is a well-knownset-theoretic bijection of Aut( X ∗ ) with Sym( X ) X ∗ sending an element g toits ‘portrait’ ( π ( g w )) w ∈ X ∗ ∈ Sym( X ) X ∗ . Put Y := X \ O . An element g ∈ Aut( X ∗ ) belongs to G if and only if g w = 1 for w / ∈ O ∗ and π ( g w ) ∈ Sym( Y ) for w ∈ O ∗ (where Sym( Y ) is viewed as a subgroup of Sym( X )in the usual way). There is thus a bijection π : G → Sym( Y ) O ∗ given by π ( g ) = ( π ( g w )) w ∈ O ∗ . The mapping π is a homomorphism because if f, g ∈ G and w ∈ O ∗ , then ( f g ) w = f g ( w ) g w = f w g w as g fixes O ∗ pointwise. Thisproves the first statement. Since the variety of groups generated by anyfinite group is locally finite, it follows that Sym( Y ) O ∗ is locally finite. (cid:3) Lemma 7.5.
Suppose that G = h Y i ≤ Aut( X ∗ ) is a generalized basilicagroup generated by directed elements. If there is an orbit O ⊆ X of G suchthat v g ∈ O for all g ∈ Y , then the homomorphism φ : G → Aut( O ∗ ) inducedby the action G y O ∗ enjoys the following properties:(1) φ ( G ) is a self-replicating generalized basilica group with generatingset φ ( Y ) , and(2) ker( φ ) is locally finite.Proof. For (1), Lemma 7.3 ensures that φ ( G ) ≤ Aut( O ∗ ) is a generalizedbasilica group with generating set φ ( Y ). Lemma 5.3 implies that sec G ( v ) = G for all v ∈ O , so sec φ ( G ) ( v ) = φ ( G ) for all v ∈ O . Since G acts transitivelyon O , φ ( G ) is self-replicating.For (2), clearly ker( φ ) fixes O ∗ pointwise. Since v g ∈ O for all g ∈ Y ,it follows that f v = 1 for all v ∈ X \ O and f ∈ G . As G is self-similar,it follows that f w = 1 for all f ∈ G and w / ∈ O ∗ . Indeed, we may write w = uav with u ∈ O ∗ , a ∈ X \ O , and v ∈ X ∗ , so f w = (( f u ) a ) v = 1 v = 1.Thus, ker( φ ) is isomorphic to a subgroup of Sym( X \ O ) O ∗ , and hence islocally finite, by Lemma 7.4. (cid:3) Bringing together our lemmas produces the desired reduction theorem.To state our main theorem in full generality requires a definition.
Definition 7.6.
Suppose that Q is a property of generalized basilica groups.We say that Q is a sub-basilica stable property if for every generalizedbasilica group G = h Y i ≤ Aut( X ∗ ) with property Q , the following holds: Forevery generalized basilica group of the form h Y ′ i ≤ Aut( X ∗ ) with Y ′ ⊂ Y ,any generalized basilica group g h Y ′ i induced by a sub-basilica action of h Y ′ i has property Q . In particular, h Y ′ i has Q .Observe that any property of groups stable under passing to subgroupsand quotients is sub-basilica stable. Theorem 7.7.
Let Q be a sub-basilica stable property and suppose that P isa property of groups enjoyed by abelian groups and stable under taking sub-groups, finite direct products, and P -by-finite groups. If every self-replicatinggeneralized basilica group with property Q has property P , then every gen-eralized basilica group with property Q is (locally finite)-by- P .Proof. Suppose that G = h Y i ≤ Aut( X ∗ ) is a generalized basilica group withproperty Q . We argue by induction on the number of directed elements in Y ,denoted by δ ( Y ), for the theorem. If δ ( Y ) ≤
1, then G is virtually abelianby Lemma 5.6, and we are done. Suppose the theorem holds up to n andsay that δ ( Y ) = n + 1. Let O , . . . , O l list the orbits of G on X and let Y i ⊆ Y be the collection of directed d ∈ Y such that v d ∈ O i . Lemma 7.2supplies an injection G ֒ → l Y i =1 Sym( O i ) ⋉ l Y i =1 H O i i . It is enough to show that each H i is (locally finite)-by- P Each H i ≤ Aut( X ∗ ) is again a generalized basilica group with property Q , since Q is sub-basilica stable. If no O i contains v d for every directed d ∈ Y , each H i is such that δ ( Y i ) < n + 1. The inductive hypothesis thenensures that each H i is (locally finite)-by- P .Suppose some O i contains v d for every directed d ∈ Y . Without lossof generality, i = 1, so H j = { } for j >
1. Clearly each H j with j > P . We apply Lemma 7.5 to the group H . Letting K := ker( H y O ∗ ), Lemma 7.5 ensures that H /K ≤ Aut( O ∗ ) is a self-replicating generalized basilica group, and K is locally finite. Since H /K is the group induced by a sub-basilica action, H /K has property Q , hence H /K has property P . We conclude that H is (locally finite)-by- P . (cid:3) We can do better for torsion-free bounded automata groups. The proofis essentially the same as that of Theorem 7.7, so we leave it to the reader.
Theorem 7.8.
Suppose that P is a property of groups enjoyed by abeliangroups and stable under taking subgroups, finite direct products, and P -by-finite groups. If every self-replicating torsion-free bounded automata grouphas property P , then every torsion-free bounded automata group has P . N ELEMENTARY AMENABLE BOUNDED AUTOMATA GROUPS 31
Groups with center. A system of imprimitivity for a group action G y X is a G -equivariant equivalence relation on X . That is, x ∼ y if andonly if g ( x ) ∼ g ( y ). The equivalence classes of a system of imprimitivity arecalled blocks of imprimitivity . A block B is called trivial if it equalseither X or a singleton set. Lemma 7.9.
Suppose that h ∈ Aut( X ∗ ) is a directed automorphism suchthat h x ∈ { , h } for all x ∈ X . Let B ⊆ X be a non-trivial block of imprim-itivity for the action of h h i on X and O be the orbit of v h under the actionof h h i on X . If | O ∩ B | ≥ , then there is j ∈ Z and w ∈ B such that h j stabilizes B setwise, ( h j ) w = h ± , and ( h − j ) w = 1 .Proof. Let i ≥ h i ( v h ) ∈ B .Suppose first that i = 0, so v h ∈ B . Set w := v h . Let j > h j ( w ) ∈ B and observe that h j ( w ) = w since | O ∩ B | ≥
2. The element h j stabilizes B setwise, and it is clear that ( h j ) w = h . On the other hand,( h − j ) has non-trivial sections at { h ( w ) , . . . , h j ( w ) } , and this set excludes w .We conclude that ( h − j ) w = 1, establishing the lemma in this case.Suppose next that i > w := h i ( v h ). Let j > h j + i ( v h ) ∈ B . The element h − j setwise stabilizes the block B , so h − j ( w ) = h i − j ( v h ) ∈ B . If j ≤ i , then we contradict the choice of i , so j > i .It now follows that ( h − j ) w = h − . On the other hand, h j ( w ) = h j + i ( v h ). As j + i > j , h − j has a trivial section at h j ( w ). We deduce that ( h j ) w = 1. (cid:3) Lemma 7.10.
Let G ≤ Aut( X ∗ ) be self-similar, = d ∈ G be directed suchthat d x ∈ { , d } for all x ∈ X , and O be the orbit of v d ∈ X under theaction of h d i . Then, O is setwise stabilized by C G ( d ) , and g v ∈ C G ( d ) forall g ∈ C G ( d ) and v ∈ O .Proof. Take n ≥ d n fixes O . The element d n is such that( d n ) v = ( d if v ∈ O v ∈ O and z ∈ C G ( d ), we see that( d − n ) z ( v ) z v d = ( d − n zd n ) v = z v . The sections of d − n are either d − or 1, and the non-trivial sections occurfor exactly the v ∈ O . It is therefore the case that z ( v ) ∈ O and d commuteswith z v . The group C G ( d ) thus setwise stabilizes O , and g v ∈ C G ( d ) for all g ∈ C G ( d ) and v ∈ O . (cid:3) Lemma 7.11.
For G ≤ Aut( X ∗ ) , if v ∈ X ∗ is such that sec G ( v ) = G , then sec Z ( G ) ( v ) ≤ Z ( G ) . In particular, if G is weakly self-replicating, then Z ( G ) is weakly self-similar.Proof. We see that Z ( G ) ( v ) ≤ Z ( G ( v ) ) ≤ G ( v ) . Therefore,sec Z ( G ) ( v ) = φ v ( Z ( G ) ( v ) ) ≤ φ v ( Z ( G ( v ) )) ≤ Z ( φ v ( G ( v ) )) = Z ( G ) . The lemma now follows. (cid:3)
Theorem 7.12.
Suppose that G = h Y i ≤ Aut( X ∗ ) is a self-replicatinggeneralized basilica group. If Z ( G ) is non-trivial, then G is abelian.Proof. By Proposition 5.4, we may assume that Y consists of directed ele-ments. Set Z := Z ( G ) and for O ⊆ X , define S O := { z v | z ∈ Z and v ∈ O } ∪ Z. Fix d ∈ Y and let O d be the orbit of h d i on X such that v d ∈ O d .Lemma 7.10 ensures that C G ( d ) contains S O d , so d ∈ C G ( S O d ). Let W ⊆ X be maximal such that O d ⊆ W and d ∈ C G ( S W ). We argue that W = X .Let us first see that W is invariant under the action of Z . Take v ∈ W and z ∈ Z . For g ∈ Z , we see that ( gz ) v = g z ( v ) z v . Since d commutes with( gz ) v and z v , it follows that d commutes with g z ( v ) . The element d thuscommutes with g z ( v ) for every g ∈ Z , hence z ( v ) ∈ W , since W is maximal.We next argue that W is invariant under each c ∈ Y . Fixing c ∈ Y , wehave two cases. For the first case, suppose that O c ∩ W = ∅ , where O c ⊆ X is the orbit of v c under h c i . Take g ∈ Z and fix v ∈ W . We see that g v = ( c − gc ) v = ( c − ) gc ( v ) g c ( v ) c v = ( c − ) cg ( v ) g c ( v ) Since W is invariant under Z , g ( v ) ∈ W . The element cg ( v ) is thus not in O c , so ( c − ) cg ( v ) = 1. We conclude that g v = g c ( v ) , hence d commutes with g c ( v ) . Therefore, c ( v ) ∈ W by maximality, and W is invariant under theaction of c .For the second case, suppose that O c ∩ W = ∅ . Applying Lemma 7.10, O c is Z -invariant. The intersection O c ∩ W is then also Z -invariant, so | O c ∩ W | ≥
2, since Z cannot fix any vertex of X by Lemmas 4.6 and 7.11.The orbits of Z on O c form a non-trivial system of imprimitivity for h c i , sothere is a non-trivial block of imprimitivity B ⊆ O c ∩ W for the action of h c i on X . Via Lemma 7.9, we may find w ∈ B and j ∈ Z such that ( c j ) w = c ± ,( c − j ) w = 1, and c j setwise fixes B .Let z ∈ Z be such that z ( c j ( w )) = w . The element c − j zc j is an elementof Z , and w ∈ W . The element d thus commutes with ( c − j zc j ) w . On theother hand, ( c − j zc j ) w = ( c − j ) w z c j ( w ) ( c j ) w = z c j ( w ) c ± , so d commutes with z c j ( w ) c ± . Furthermore, d commutes with z c j ( w ) since c j ( w ) ∈ B ⊆ W . We deduce that d commutes with c .For any vertex u ∈ W and g ∈ Z , d commutes with ( c − gc ) u . On theother hand, ( c − gc ) u = ( c − ) g ( c ( u )) g c ( u ) c u Since the sections of c are either equal c or trivial, d commutes with c u and( c − ) g ( c ( u )) . We conclude that d commutes with g c ( u ) , so c ( u ) ∈ W by thechoice of W . The set W is thus setwise fixed by every c ∈ Y . Our group G = h Y i acts transitively on X , so it is the case that W = X .For any c ∈ Y , there is z ∈ Z such that z ( c ( v c )) = c ( v c ) by Lemma 4.6.Taking such a z , c − zc ∈ Z , and we have that d commutes with ( c − zc ) v c , N ELEMENTARY AMENABLE BOUNDED AUTOMATA GROUPS 33 by our work above. Furthermore,( c − zc ) v c = z c ( v c ) c, and d commutes with z c ( v c ) . We infer that d commutes with c .Since d is arbitrary, it now follows that all elements of Y commute, hence G is abelian. (cid:3) We can now upgrade Theorem 5.8 in the self-replicating case. We notethat Theorem 5.8 cannot be upgraded itself since one can always add finitaryelements.
Corollary 7.13.
Suppose that G = h Y i ≤ Aut( X ∗ ) is a self-replicatinggeneralized basilica group. If G contains an odometer, then either G isabelian or G is not elementary amenable.Proof. Suppose that G is elementary amenable. By Theorem 5.8, G is vir-tually abelian. Suppose for contradiction that G is not abelian. Apply-ing Corollary 4.4, there is H E G such that H is weakly self-replicating,rk( H ) + 1 = rk( G ), and G/H is finite. Observe that H is abelian, since itmust be infinite with rank 0.Take h ∈ G an odometer. Since G/H is finite, there is i > h i ∈ H . Set m := | X | i . The element h m is in H , and ( h m ) x = h i for all x ∈ X . Since H is abelian, f − h m f = h m for any f ∈ H . For any x ∈ X ,it is then the case that ( f x ) − h i f x = h i . We conclude that C G ( h i ) contains H along with f x for all f ∈ H and x ∈ X .Take d ∈ Y . In view of Lemma 4.8, there is f ∈ H such that f ( d ( v d ))) = d ( v d ). The element d − f d is in H , and ( d − f d ) v d = f d ( v d ) d . The centralizer C G ( h i ) therefore contains f d ( v d ) d . Since C G ( h i ) also contains f d ( v d ) , we inferthat d ∈ C G ( h i ). The element h i is therefore central in G .The group G thus has non-trivial center. Applying Theorem 7.12, weconclude that G is abelian, which is absurd. (cid:3) Theorem 7.12 and Theorem 7.7 together imply all nilpotent generalizedbasilica groups are virtually abelian.
Corollary 7.14.
Every nilpotent generalized basilica group is virtually abelian.Proof.
Nilpotent groups have non-trivial centers. Theorem 7.12 thus impliesthat every self-replicating nilpotent generalized basilica group is virtuallyabelian.The property Q of being nilpotent is closed under subgroups and quo-tients, so a fortiori it is a sub-basilica stable property. The property P ofbeing virtually abelian is enjoyed by abelian groups and stable under takingsubgroups, finite direct products, and forming P -by-finite groups. We maythus apply Theorem 7.7 to conclude that every nilpotent generalized basilicagroup is (locally finite)-by-virtually abelian.Every subgroup of a finitely generated nilpotent group is finitly gener-ated. Hence, every nilpotent generalized basilica group is finite-by-virtually abelian. Since generalized basilica groups are residually finite, it follows thatevery nilpotent generalized basilica group is virtually abelian. (cid:3) Corollary 7.15.
Every torsion-free nilpotent bounded automata group isvirtually abelian.
Balanced groups.Definition 7.16.
Let G = h Y i ≤ Aut( X ∗ ) be a generalized basilica group.We say that G is balanced if for each directed g ∈ Y , the least n ≥ g n fixes v g ∈ X is also such that g n fixes X pointwise.For G = h Y i ≤ Aut( X ∗ ) balanced, take g ∈ Y directed and let c . . . c n bethe cycle deomposition of π ( g ) ∈ Sym( X ). That G is balanced ensures thatfor c i such that v g appears in c i , the order | c j | divides | c i | for all 1 ≤ j ≤ n .This condition gives us control over the generators analogous to how thetree-like condition did for kneading automata groups. Theorem 7.17.
Suppose that G = h Y i ≤ Aut( X ∗ ) is a self-replicatinggeneralized basilica group. If G is balanced, then either G is abelian or G isnot elementary amenable.Proof. By Proposition 5.4, we may assume that Y consists of directed ele-ments.Suppose that G is elementary amenable. If Y contains an element thatacts transitively, i.e., an odometer, then Corollary 7.13 implies that G isabelian, and we are done.Let us suppose that each element of Y does not act transitively and sup-pose toward a contradiction that G is not abelian. Applying Corollary 4.4,there is H E G such that H is weakly self-replicating, rk( H ) + 1 = rk( G ),and G/H is either finite or abelian.For each d ∈ Y , let n be least such that d n fixes v d . Since G is balanced, d n fixes X , and as h d i does not at transitively on X , there is x ∈ Y suchthat ( d n ) x = 1. In view of Lemma 4.7, we deduce that d ∈ H . As d isarbitrary, we conclude that H = G which is absurd since rk( H ) < rk( G ).The group G is thus abelian. (cid:3) Corollary 7.18.
Every balanced generalized basilica group is either (locallyfinite)-by-(virtually abelian) or not elementary amenable.Proof.
Theorem 7.17 ensures that every self-replicating generalized basilicagroup that is balanced and elementary amenable is virtually abelian.The property Q of being balanced and elementary amenable is a sub-basilica stable property. The property P of being virtually abelian is enjoyedby abelian groups and stable under taking subgroups, finite direct products,and forming P -by-finite groups. We may thus apply Theorem 7.7 to concludethat every balanced and elementary amenable generalized basilica group is(locally finite)-by-virtually abelian. (cid:3) Remark 7.19.
Example 9.1 below shows that Theorem 7.18 is sharp.
N ELEMENTARY AMENABLE BOUNDED AUTOMATA GROUPS 35
We conclude this section with a sufficient condition to be a balancedgeneralized basilica group.
Proposition 7.20.
Suppose that G = h Y i ≤ Aut( X ∗ ) is a generalizedbasilica group that acts transitively on X . If π ( G ) ≤ Sym( X ) is abelian,then G is balanced.Proof. Let d ∈ Y be directed. The image π ( h d i ) is a normal subgroup of π ( G ), so the orbits of h π ( d ) i form a system of imprimitivity for the actionof π ( G ) on X . Since π ( G ) is transitive, all orbits of h π ( d ) i have the samesize, and it follows that d is balanced. (cid:3) Groups of abelian wreath type
Definition 8.1.
We say that G ≤ Aut( X ∗ ) is of abelian wreath type if π ( G ) ≤ Sym( X ) is abelian and G admits a finite self-similar generating set Y such that for every g ∈ Y either g x = 1 for all x ∈ X or g x ∈ { g }∪ π ( G ) forall x ∈ X with exactly one x such that g x = g . We call Y a distinguishedgenerating set and write G = h Y i ≤ Aut( X ∗ ) to indicate that Y is adistinguished generating set for G .A straightforward verification shows that a group of abelian wreath typeis a bounded automata group. Let us note several immediate consequencesof abelian wreath type; the proofs are elementary and so are left to thereader. Observation 8.2.
Let G ≤ Aut( X ∗ ) be a self-replicating group of abelianwreath type.(1) For all non-trivial f ∈ Sym( X ) ∩ G and v ∈ X , f ( v ) = v .(2) For all v ∈ X , G ( v ) = G ( X ) .(3) For all g, h ∈ G , [ g, h ] ∈ G ( X ) The case of the binary tree.
We here consider the bounded au-tomata groups G that have a faithful representation in reduced form on[2] ∗ . Any faithful representation G = h Y i ≤ Aut([2] ∗ ) in reduced form isnecessarily of abelian wreath type.Let us first list all possible non-trivial automorphisms of [2] ∗ that canappear as the distinguished generators of a bounded automata group G = h Y i ≤ Aut([2] ∗ ) in reduced form. We give the automorphisms in wreathrecursion. Observation 8.3.
Letting σ ∈ Sym(2) be the non-trivial element, any non-trivial generator g of a bounded automata group G = h Y i ≤ Aut([2] ∗ ) inreduced form has one of the following types, up to taking an inverse:(I) g = σ ,(II) g = ( σ, g ) or g = ( g, σ ) ,(III) g = σ (1 , g ) , or(IV) g = σ ( σ, g ) . Lemma 8.4. If G = h Y i ≤ Aut([2] ∗ ) is a bounded automata group inreduced form and G contains an element of type ( IV ) , then G is not ele-mentary amenable.Proof. The group G must contain σ , since it is self similar and contains anelement of type ( IV ). Say that g = σ ( σ, g ) is an element of type ( IV ) in G .It suffices to show H := h σ, g i is not elementary amenable.Let us suppose toward a contradiction that H is elementary amenable.The group H is a self-replicating and clearly non-abelian, so we apply Corol-lary 4.4 to find a weakly self-replicating M E G such that rk( M )+1 = rk( G ).In view of Lemma 4.6, we infer that M acts transitively on [2].We see that σg = ( σ, g ), so ( σg ) = (1 , g ). Taking m ∈ M such that m (0) = 1, we have that [(1 , g ) , m ] ∈ M , and as M is weakly self-similar,[(1 , g ) , m ] = g ∈ M . The square g equals ( gσ, σg ), so σg = ( σ, g ) ∈ M ,using again that M is weakly self-similar. A final application of weak self-similarity implies that σ and g are elements of M . This is absurd, since M must be a proper subgroup. (cid:3) Lemma 8.5. If G ≤ Aut([2] ∗ ) is a bounded automata group of abelianwreath type and G contains elements of type ( II ) and ( III ) , then G is notelementary amenable.Proof. The group G must contain σ , since it is self similar and containsan element of type ( II ). As the proofs are the same, let us assume that G contains g = ( σ, g ). Let us also assume that the type ( III ) elementis h = σ (1 , h ). It now suffices to show H := h σ, g, h i is not elementaryamenable.Let us suppose toward a contradiction that H is elementary amenable.The group H is a self-replicating and clearly non-abelian, so we apply Corol-lary 4.4 to find a weakly self-replicating M E G such that rk( M )+1 = rk( G ).In view of Lemma 4.6, we infer that M acts transitively on [2].We see that σh = (1 , h ). Taking m ∈ M such that m (0) = 1, we havethat [(1 , h ) , m ] ∈ M , and as M is weakly self-similar, [(1 , h ) , m ] = h ∈ M .The commutator [ g, h ] is then also an element of M . Moreover,[ g, h ] = ghg − h − = ( σ, g ) σ (1 , h )( σ, g − ) σ ( h − ,
1) = ( σhg − h − , gσ )we conclude that gσ ∈ M , since M is weakly self-replicating. As h = σ (1 , h ) ∈ M , it is then the case that gσh = ( σ, g )(1 , h ) = ( σ, gh ) ∈ M . Froma second application of weak self-similarity, it follows that σ , g , and h areelements of M . This is absurd as M is a proper subgroup of H . (cid:3) Remark 8.6.
The groups H arising in the proofs of Lemma 8.4 and 8.5 seemlike they may be of independent interest. They appear to be the “smallest”non-elementary amenable bounded automata groups. Theorem 8.7. If G = h Y i ≤ Aut([2] ∗ ) is a bounded automata group inreduced form, then either G is virtually abelian or G is not elementaryamenable. N ELEMENTARY AMENABLE BOUNDED AUTOMATA GROUPS 37
Proof.
Suppose that G is elementary amenable. In view of Lemma 8.4, Y contains no element of type ( IV ). We now have two cases.Suppose first that Y contains one or two elements of type ( II ). Thecase of one element is an easy adaptation of the proof for the case of twoelements, so we only consider the latter. In view of Lemma 8.5, Y does notcontain an element of type ( III ). It now follows that Y has three non-trivialelements: Y = { σ, ( g, σ ) , ( σ, h ) } . Seeing as σ ( σ, g ) σ = ( h, σ ), the group G is generated by two involutions, σ, ( σ, g ), so G is either finite or the infinitedihedral group.Suppose next that Y contains an element of type ( III ). As in the previouscase, Y does not contain an element of type ( II ). The non-trivial elements of Y then consist of either a single type ( III ) element or a type (
III ) elementand a type ( I ) element. In the former case, G is abelian, and in the lattercase, G is virtually abelian. (cid:3) Elementary amenable groups.
We are now prepared to prove ourmain theorem of this section.
Lemma 8.8.
Suppose that G = h Y i ≤ Aut( X ∗ ) is a self-replicating groupof abelian wreath type that is elementary amenable. If G is non-abelian, thenthere is some directed d ∈ Y and x ∈ X such that d x is a non-trivial finitaryelement.Proof. We prove the contrapositive. If every directed d ∈ Y is such that d x ∈ { , d } for all x ∈ X , then G is a generalized basilica group, and in viewof Proposition 7.20, G is furthermore balanced. Theorem 7.17 thus impliesthat G is abelian. (cid:3) Suppose that G = h Y i ≤ Aut( X ∗ ) is a self-replicating group of abelianwreath type with | X | > G is elementary amenablebut not virtually abelian. Corollary 5.11 supplies a weakly self-replicating M E G such that rk( M ) + 1 = rk( G ) and G/M is finite. Recall that M acts transitively on X by Lemma 4.8. Setting F := π ( G ) ∩ G , Lemma 8.12below claims that G = F M ; by Lemma 8.8, F contains non-trivial elements.To prove this requires three preliminary lemmas, that amount to checkingthree cases.Fix c ∈ Y some directed element, let v ∈ X be the active vertex of c on X , and say that O ⊆ X is the orbit of v under the action of h c i on X . Lemma 8.9.
If there is l ∈ F such that lc fixes some x ∈ X , then there is k ∈ F such that kc ∈ M .Proof. Set d := lc . Since G has abelian wreath structure, d fixes X . If d w = 1 for some w ∈ X , then c ∈ M by Lemma 4.7, since d v = c . We thussuppose that d w = 1 for all w ∈ X . In view of Lemma 8.8, we may fix a non-trivial f ∈ F , and we may also fix m ∈ M such that m ( v ) ∈ X \ { v, f ( v ) } .Such an m exists since | X | > M is weakly self-replicating and of finiteindex. Set y := m v . The commutator [ m − , d − ] is in M , so[ m − , d − ] v = ( m − d − md ) v = y − hyc is in M for some h ∈ F . On the other hand, [ m − , f d − f − ] is also anelement of M , so [ m − , f d − f − ] v = y − h ′ yd f − ( v ) is in M for some h ′ ∈ F . The element k := d f − ( v ) is also a non-trivialfinitary element.We now have that c and a non-trivial finitary element k are elements of y − F yM . The quotient y − F yM/M is abelian, so [ k, c − ] ∈ M . As k − moves c ( v ), it follows that [ k, c − ] v = k ′ c for some k ′ ∈ F . Thus, k ′ c ∈ M ,completing the proof. (cid:3) Lemma 8.10.
If some non-trivial element of F setwise stabilizes O , thenthere is k ∈ F such that kc ∈ M .Proof. Fix a non-trivial f ∈ F such that f stabilizes O . If | O | ≤
2, then f c fixes pointwise O , and Lemma 8.9 implies that there is k ∈ F such that kc ∈ M . We thus assume that | O | > j > c j fixes O and take i < j least such that f c i fixes O pointwise. If i = j −
1, then f c − fixes O pointwise, and thereis k ∈ F such that kc − ∈ M by Lemma 8.9. In this case, ck − ∈ M ,so conjugating by k − , we see that k − c ∈ M . We may thus assume that i < j −
1. If f c i has a trivial section at some z ∈ X , then as in the previouslemma, Lemma 4.7 implies that kc ∈ M for some k ∈ F . We may thus alsoassume that all sections of f c i at z ∈ X are non-trivial.The element f c i has non-finitary sections atΩ := { v, c − ( v ) , . . . , c − i +1 ( v ) } . Note that | O \ Ω | ≥
2, since | Ω | = i while | O | = j . We may find m ∈ M such that m ( v ) ∈ O \ (Ω ∪ c − (Ω)) , since Ω ∪ c − (Ω) is only one element larger than Ω. Set y := m v .The commutator [ m − , ( f c i ) − ] is in M , so[ m − , ( f c i ) − ] v = y − h yh c is in M for some h , h ∈ F . On the other hand, c − f c i +1 has non-finitarysections exactly at Ω ∪ c − (Ω). In particular, ( c − f c i +1 ) v = c − k c for somenon-trivial k ∈ F , since we assume that f c i has no trivial sections. Thecommutator [ m − , ( c − f c i +1 ) − ] is in M , so[ m − , ( c − f c i +1 ) − ] v = y − k yc − k c is in M , for some k ∈ F .We now have that h c and c − k c are elements of y − F yM , hence h c and k are elements of y − F yM , since F is an abelian group. The quotient y − F yM/M is abelian, so [ k , ( h c ) − ] ∈ M . As k − moves c ( v ), it follows N ELEMENTARY AMENABLE BOUNDED AUTOMATA GROUPS 39 that [ k , ( h c ) − ] v = k ′ c where k ′ ∈ F . Thus, k ′ c ∈ M , completing theproof. (cid:3) Lemma 8.11. If c strongly active and no non-trivial element of F setwisestabilizes O , then there is k ∈ F such that kc ∈ M .Proof. Suppose first that h c i has only two orbits O = O and O on X . Thegroup F therefore has exponent two. Take i > c i fixes O .Let m ∈ M be such that m ( v ) ∈ O and set y := m v . We now see that[ m − , c − i ] v = y − h yhc is in M , where h , h ∈ F . The group M is normal,so hcy − h y ∈ M . Hence, hcy − h yy − h yhc = ( hc ) ∈ M. If h = 1, then considering the section [ f, c − ] v for f ∈ F non-trivial showsthat kc ∈ M for some k ∈ F , since f − ( O ) = O and c is not trivial. Letus thus suppose that h is non-trivial, so h ( O ) = O . The element ( hc ) hasnon-finitary sections exactly at v and c − h ( v ). The inverse ( hc ) − thereforehas non-finitary sections exactly at c ( v ) and ch ( v ). We compute:[ c − , ( hc ) − ] c − h ( v ) = ( c − ) h ( v ) (( hc ) − ) c h ( v ) c ch ( v ) (( hc ) ) c − h ( v ) . It now follows that [ c − , ( hc ) − ] c − h ( v ) = l cl for l , l ∈ F . Conjugating by l , we see that there is k ∈ F such that kc ∈ M .Let us suppose finally that h c i has at least three orbits O , O , and O and let i be least such that c i fixes O . If c i has a trivial section, thenLemma 4.7 ensures that kc ∈ M for some k ∈ F . We may thus assume thatevery section of c i is non-trivial. Fix f ∈ F non-trivial, and without lossof generality, f ( O ) = O . Let m ∈ M be such that m ( v ) ∈ O and set y := m v . The commutator [ m − , c − i ] is in M , so[ m − , c − i ] v = y − h yh c is in M , where h , h ∈ F . The commutator [ m − , f c − i f − ] is also in M , so[ m − , f c − i f − ] v = y − k yk is in M where k , k ∈ F , since f c i f − has non-finitary sections only on O .Note also that k is non-trivial since c i has every section non-trivial.We now have that h c and a non-trivial finitary element k are elementsof y − F yM . The quotient y − F yM/M is abelian, so [ k , ( h c ) − ] ∈ M .As k moves v , it follows that [ k , ( h c ) − ] v = k ′ c where k ′ ∈ h F i . Thus, k ′ c ∈ M . (cid:3) Bringing together the previous three lemmas, we obtain the desired result.
Lemma 8.12.
Suppose that G = h Y i ≤ Aut( X ∗ ) with | X | > is a self-replicating group of abelian wreath type. If G is elementary amenable butnot virtually abelian, then the weakly self-replicating M E G supplied byCorollary 5.11 is such that ( π ( G ) ∩ G ) M = G and rk( M ) + 1 = rk( G ) . Inparticular, G/M is finite and abelian.
Proof.
Let M E G be the finite index subgroup given by Corollary 5.11 andtake d ∈ Y directed. If d is not strongly active, then f := 1 ∈ F := π ( G ) ∩ G is such that f d fixes a vertex in X . Lemma 8.9 thus gives k ∈ F such that kd ∈ M . If d is strongly active, then either there is some non-trivial f ∈ F such that f setwise stabilizes the orbit of v d ∈ X under the action of h d i acting on X or there is not. In the former case, Lemma 8.10 supplies k ∈ F such that kd ∈ M and in the latter, Lemma 8.11 gives k ∈ F such that kd ∈ M .It now follows that F M = G , and the lemma is verified. (cid:3) The desired theorem is now in hand.
Theorem 8.13. If G = h Y i ≤ Aut( X ∗ ) is a self-replicating group ofabelian wreath type, then either G is virtually abelian, or G is not elementaryamenable.Proof. The case that | X | = 2 is settled by Theorem 8.7. We thus assumethat | X | >
2. Set F := π ( G ) ∩ G . We suppose toward a contradiction that G is elementary amenable but not virtually abelian.Applying Corollary 5.11, we obtain non-trivial weakly self-replicating L E G and M E G such that L ≤ M , G/M is finite,
M/L is abelian, andrk( L ) + 2 = rk( M ) + 1 = rk( G ). Lemma 8.12 ensures further that F M = G .We now argue that G/L is abelian, which contradicts the rank of G .We have two cases: (1) M ∩ F = { } , and (2) M ∩ F = { } . For case(1), fix a non-trivial f ∈ F ∩ M . By Lemma 8.12, each d ∈ Y directedadmits h ∈ F such that hd ∈ M . Taking hd ∈ M , [ f, ( hd ) − ] ∈ L , so[ f, ( hd ) − ] v d = kd is an element of L where k ∈ F . We conclude that foreach directed d ∈ Y there is h ∈ F such that hd ∈ L . It is thus the casethat F L = G , so G/L is abelian, giving the desired contradiction.For case (2), fix d ∈ Y directed and let h ∈ F be such that hd ∈ M . If hd fixes v d , then ( hd ) v d = d ∈ M . Hence h ∈ F ∩ M = { } . The element d must be strongly active since else since d = 1, it has a finitary section thatbelongs to M and so M contains a non-trivial element of F . We concludethat d does not fix v d . We may thus assume that hd does not fix v d . Set˜ d := hd and observe that ˜ d has exactly one non-finitary section, namely˜ d v d = d .If h ˜ d i acts intransitively on X , let i be least such that ˜ d i fixes X . Since M intersects F trivially, it must be the case that ˜ d i has trivial sections atall w outside the orbit of v d under h ˜ d i . The group M acts transitively on X , so Lemma 4.7 implies that h ′ d ∈ L for some h ′ ∈ F .Suppose that h ˜ d i acts transitively on X . Fix f ∈ F that is non-trivial;such an element exists by Lemma 8.8. If f ( v d ) = ˜ d ( v d ), then f acts likea | X | -cycle on X , since π ( G ) is abelian. Since | X | > f is non-trivial,and f ( v d ) = ˜ d ( v d ). By possibly replacing f with f , we may assume that f ( v d ) = ˜ d ( v d ). The elements f ˜ d − f − and ˜ d are in M , so [ f ˜ d − f − , ˜ d − ] ∈ N ELEMENTARY AMENABLE BOUNDED AUTOMATA GROUPS 41 L . Moreover,[ f ˜ d − f − , ˜ d − ] v d = ( ˜ d − ) f − ˜ d ( v d ) ( ˜ d − ) ˜ d ( v d ) ˜ d f − ˜ d ( v d ) d = h ′ d, where h ′ ∈ F .We now conclude that for all d ∈ D , there is h ∈ F such that hd ∈ L .Hence, F L = G , and so G/L is abelian, giving the desired contradiction. (cid:3)
The infinite dihedral group shows that virtually abelian groups indeedarise as self-replicating groups of abelian wreath type; see Example 9.2.8.3.
GGS groups.
We pause here to recall the GGS groups and observethat Theorem 8.13 applies to these groups. See [3] for more information onthese groups.For p an odd prime, form the p -regular rooted tree [ p ] ∗ . Let a ∈ Aut([ p ] ∗ )be such that a ( iz ) = (( i + 1) mod p ) z for word iz ∈ [ p ] ∗ , where i ∈ [ p ].The automorphism a cyclically permutes the first level of the tree and actsrigidly below the first level. Take a p − α := ( e , . . . , e p − ) where e i ∈ { , . . . , p − } for each 0 ≤ i ≤ p −
2. Using wreath recursion, we define b α ∈ Aut([ p ] ∗ ) by b α := ( a e , . . . , a e p − , b ). Definition 8.14.
For α = ( e , . . . , e p − ) a p − e i ∈ { , . . . , p − } for each 1 ≤ i ≤ p −
1, the
GGS group associated to α is the group G α := h a, b α i ≤ Aut([ p ] ∗ ). Proposition 8.15.
If some coordinate of α is non-zero, then the GGS group G α is a self-replicating group of abelian wreath type.Proof. We see that π ( G α ) = h a i = C p . Thus, G α is of abelian wreath type.That G α is self-replicating is also clear, and well-known. (cid:3) Corollary 8.16.
The only elementary amenable GGS groups are the cyclicgroups C p . Examples
Balanced generalized basilica groups.
Define a, b ∈ Aut([5] ∗ ) bywreath recursion: a := (01)(34)(1 , , , , a ) and b := (12)(34)(1 , , , b, . The group G := h a, b i ≤ Aut([5] ∗ ] is a balanced generalized basilica group.On [5], G has two orbits U := { , , } and V := { , } , and v a , v b areelements of V .Setting c := ab , an easy computation shows that c = (012)(1 , , , c, c fixes V , and it follows that c ∈ K := ker( G y V ∗ ). In viewof Lemma 7.5, K is locally finite, and since G is two generated, it followsthat G/K ≃ Z . The group G is thus elementary amenable. Claim.
The set { b − n cb n | n ≥ } consists of pairwise distinct elements of K . Proof.
Noting that b = (1 , , , b, b ), it is immediate that b − n cb n = (012)(1 , , , b − n cb n , . An easy calculation shows that b − n − cb n +1 = (021)(1 , , , , b − n cb n ) . Arguing by induction on i for the claim that { b − n cb n | i ≥ n ≥ } consistsof pairwise distinct elements proves the claim. (cid:3) The claim implies the kernel K is infinite. Since all subgroups of a finitelygenerated virtually abelian group are finitely generated, we deduce that G is not virtually abelian. This example shows that Corollary 7.18 is sharp.We next exhibit a large family of generalized basilica groups. Definition 9.1.
For a prime p ≥
3, let c be the cycle (0 . . . p −
1) and define g p , h p ∈ Aut([ p ] ∗ ) by g p := c ( g p , , . . . ,
1) and h p := c (1 , h p , , . . . , G p := h g p , h p i . Proposition 9.2.
For each prime p ≥ , G p is a self-replicating balancedgeneralized basilica group, weakly branch, and not elementary amenable.Proof. It is immediate that G := G p is a balanced generalized basilica group,since g := g p and h := h p act transitively on [ p ]. The elements g p and h p are such that ( g p ) x = g and ( h p ) x = h for all x ∈ X . It follows that G p is self-replicating. In view of Corollary 7.13, we need only to check that G is not abelian to conclude that G is not elementary amenable. This isimmediate by considering sections:( gh ) = 1 and ( hg ) = hg. It is clear that hg is non-trivial, so g and h do not commmute.Let us finally verify that G is weakly branch. The commutator [ g p , h ]fixes X and [ g p , h ] i = ( [ g, h ] i = 21 elseThe rigid stabilizer rist G (2) is non-trivial, and it follows that rist G ( i ) isnon-trivial for all i ∈ X . One verifies that [ g p i , h ] lies in rist G (2 i ) and[ g p i , h ] i = [ g, h ], where 2 i is the word in [ p ] ∗ consisting of i many 2’s. Weinfer that all rigid stabilizers are infinite. Since G acts transitively on eachlevel, G is weakly branch. (cid:3) To distinguish the G p requires a result from the literature on weaklybranch groups. For a tree X ∗ , the boundary of X ∗ , denoted by ∂X ∗ , isthe collection of infinite words X ω with the product topology. The groupAut( X ∗ ) has an action on ∂X ∗ by homeomorphisms which is induced fromthe action Aut( X ∗ ) y X ∗ . The product measure on ∂X ∗ arising from theuniform measure on X is preserved by Aut( X ∗ ). This measure is called the Bernoulli measure . N ELEMENTARY AMENABLE BOUNDED AUTOMATA GROUPS 43
Theorem 9.3 (See [18, Theorem 2.10.1]) . Suppose that G ≤ Aut( X ∗ ) and H ≤ Aut( Y ∗ ) are weakly branch groups. If φ : G → H is an isomorphism ofgroups, then there is a measure preserving homeomorphism F : ∂X ∗ → ∂Y ∗ such that φ ( g )( F ( α )) = F ( g ( α )) for all α ∈ ∂X ∗ , where ∂X ∗ and ∂Y ∗ areequipped with the respective Bernoulli measures Theorem 9.4.
For p = q with p, q ≥ , G p and G q are not isomorphic.Proof. Suppose for contradiction there are p = q such that G p is isomorphicto G q . Appealing to Theorem 9.3, there is a measure preserving homeomor-phism F : ∂ [ p ] ∗ → ∂ [ q ] ∗ .Fix O ⊆ ∂ [ p ] ∗ clopen with measure p . The image F ( O ) is a clopenset in ∂ [ q ] ∗ with measure p . We may then find disjoint basic clopen sets W , . . . , W n of ∂ [ q ] ∗ such that F ( O ) = ⊔ ni =1 W i . The sets W i are basic, sofor each 1 ≤ i ≤ n there is k i ≥ µ ( W i ) = q ki . We deduce that1 p = n X i =1 q k i . Letting m := max { k i | ≤ i ≤ n } , we see that q m p = n X i =1 q m − k i , and the right hand side is an integer. This implies that p divides q which isabsurd. (cid:3) The infinite dihedral group.Example 9.5.
Let X := [3] and via wreath recursion, define a, b ∈ Aut( X ∗ )by a := (01)(1 , , a ) and b := (02)(1 , b, G := h a, b i is aself-replicating generalized basilica group isomorphic to the infinite dihedralgroup.In regards to Question 1.1, Example 9.5 shows that virtually abelian isthe best one can hope for in the case of self-replicating generalized basilicagroups. Example 9.6.
Let X := [2] and via wreath recursion, define a, b ∈ Aut( X ∗ )by a := (01)(1 ,
1) and b := ( a, b ). The group G := h a, b i is a self-replicatingbounded automata group of abelian wreath type and is isomorphic to theinfinite dihedral group.Example 9.6 shows that Theorem 8.13 is sharp. One naturally wondersif the infinite dihedral group is the only possible non-abelian, but virtuallyabelian example. The lamplighter group.
Let X := [4] and form Aut( X ∗ ). Defineelements of Aut( X ∗ ) by a := (02)(13)(1 , , , a ) and b := (02)(1 , b, , G := h a, b i ≤ Aut( X ∗ ) is a bounded automata group. Infact G is a bounded automata group that is both a generalized basilica groupand of abelian wreath type.The element b has order 2, and a simple calculation shows the following: a n ba − n = (02)(1 , a n ba − n , ,
1) and a n +1 ba − n − = (02)(1 , , , a n ba − n )for all n ≥ Z .We now argue that C ≀ Z is isomorphic to G ; recall that C denotes Z / Z .This requires two claims. Claim.
For any n = m in Z , a n ba − n and a m ba − m commute Proof.
It suffices to show that a n ba − n and a m ba − m commute for all n, m ≥
0. Set Ω n := { a i ba − i | ≤ i ≤ n } . We argue by induction on n that a n +1 ba − n − commutes with every g ∈ Ω n . For the base case, n = 0, we see baba − = (02)(1 , b, , , , , b )= (1 , b, , b )= (02)(1 , , , b )(02)(1 , b, , aba − b, and thus b and aba − commute.Suppose the inductive claim holds for n and take a m ba − m ∈ Ω n +1 . Thecases that m is odd while n + 1 is even and m is even while n + 1 odd followas the base case. Let us now suppose that m and n + 1 are both even; thecase where they are both odd is similar. We then have that a n +2 ba − n − =(1 , a k ba − k , ,
1) where k < n + 2 and that a m ba − m = (02)(1 , a l ba − l , , l < n + 2. We see that a n +2 ba − n − a m ba − m = (1 , a k ba − k a l ba − l , , a m ba − m a n +2 ba − n − = (1 , a l ba − l a k ba − k , , . The inductive hypothesis ensures that a k ba − k a l ba − l = a l ba − l a k ba − k . There-fore, a m ba − m and a n +2 ba − n − commute, completing the induction. (cid:3) Claim.
For any distinct i , . . . , i n in Z , the product Q nj =1 a i j ba − i j is not 1. Proof.
Suppose for a contradiction that the claim is false, and let n be leastsuch that there are distinct i , . . . , i n for which g := Q nj =1 a i j ba − i j = 1. Byconjugating with an appropriate power of a , we may assume that all i j arepositive. We may also take the i , . . . , i n to be such that P nj =1 i j is least.The section of g at 1 must have the form Q sj =1 a k j ba − k j . Likewise, thesection of g at 3 must have the form Q rj =1 a l j ba − l j . These two sectionsmust also be trivial, and since n ≥
2, it cannot be the case that r = 0 = s . We may thus assume without loss of generality that s ≥
1. Since wechoose n to be least, it is indeed the case that s = n . However, l j < i j , so N ELEMENTARY AMENABLE BOUNDED AUTOMATA GROUPS 45 P nj =1 l j < P nj =1 i j . This contradicts out choice of i , . . . , i j . This completesthe reductio. (cid:3) Proposition 9.7.
The lamplighter group C ≀ Z is isomorphic to G .Proof. We have that C ≀ Z = L Z C ⋊ Z . We define Φ : L Z C ⋊ Z → G by( f, n ) Y i ∈ Supp( f ) a i ba − i a n where Supp( f ) := { i ∈ Z | f ( i ) = 0 } .Let us verify Φ is a homomorphism:Φ(( f, n )( g, m )) = Φ(( f + n.g, n + m ))= (cid:16)Q i ∈ Supp( f + n.g ) a i ba − i (cid:17) a n + m = (cid:16)Q i ∈ Supp( f ) a i ba − i Q i ∈ Supp( n.g ) a i ba − i (cid:17) a n + m = (cid:16)Q i ∈ Supp( f ) a i ba − i Q i ∈ Supp( g ) a i + n ba − i − n (cid:17) a n + m = (cid:16)Q i ∈ Supp( f ) a i ba − i (cid:17) a n (cid:16)Q i ∈ Supp( g ) a i ba − i (cid:17) a m = Φ(( f, n ))Φ(( g, m )) . The third equality uses that the conjugates of b by powers of a commuteand that b has order two.It is clear that Φ is surjective. For injectivity, if (cid:16)Q i ∈ Supp( f ) a i ba − i (cid:17) a n =1, then n = 0, and Q i ∈ Supp( f ) a i ba − i = 1. The second claim ensures that Q i ∈ Supp( f ) a i ba − i = 1 exactly when the product is trivial. It follows that Φis injective. (cid:3) Weak self-replication.
Let X := [2] and form Aut( X ∗ ). Let σ bethe cycle (01) and define a := σ ( σ, σa ). The group G := h a i is isomorphicto Z , so it cannot contain the section σ . On the other hand, one checksthat σaσ = a − , so a = ( a, a − ). The section homomorphism φ x : G ( x ) → Aut( X ∗ ) thus has image G . The group G is weakly self-replicating but notself-similar, so it is not self-replicating. References
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