Engel elements in weakly branch groups
Gustavo A. Fernández-Alcober, Marialaura Noce, Gareth M. Tracey
EENGEL ELEMENTS IN WEAKLY BRANCH GROUPS
GUSTAVO A. FERN ´ANDEZ-ALCOBER, MARIALAURA NOCE,AND GARETH M. TRACEY
Abstract.
We study properties of Engel elements in weakly branch groups,lying in the group of automorphisms of a spherically homogeneous rootedtree. More precisely, we prove that the set of bounded left Engel elementsis always trivial in weakly branch groups. In the case of branch groups,the existence of non-trivial left Engel elements implies that these are all p -elements and that the group is virtually a p -group (and so periodic) for someprime p . We also show that the set of right Engel elements of a weakly branchgroup is trivial under a relatively mild condition. Also, we apply these resultsto well-known families of weakly branch groups, like the multi-GGS groups. Introduction
A rapidly developing area of group theory studies the properties of branchgroups , a special kind of groups acting on spherically homogeneous rooted trees.These groups, which were first defined by Grigorchuk at the Groups St An-drews conference in Bath in 1997, are generalizations of the famous p -groupsconstructed by Grigorchuk himself [9], and by Gupta and Sidki [13]. Despitetheir relatively recent introduction, branch groups have appeared in the liter-ature in the past, without being explicitly defined. For instance, the class ofbranch groups contains one of the three classes of groups in John Wilson’s fa-mous characterisation of just infinite groups [18]. This is one of the primaryreasons for their study. Another important motivation for studying branchgroups comes from the remarkable properties that some of these groups canpossess, like intermediate growth, amenability, the congruence subgroup prop-erty, or providing a negative answer to the General Burnside Problem. In thissetting, one can also consider the larger family of weakly branch groups , whichpreserve many of the most interesting features enjoyed by branch groups. Werefer the reader to Section 2 for a quick introduction to these classes of groups.With this motivation in mind, the purpose of this paper is to investigate Engelelements in weakly branch groups. Given two elements g and x in a group G ,we define [ g, n x ] for all n ∈ N ∪ { } by means of [ g, x ] = g and, for n ≥ g, n x ] = [[ g, n − x ] , x ] = [[ . . . [[ g, x ] , x ] , . . . ] , x ] , (where x appears n times). Mathematics Subject Classification.
Key words and phrases.
Branch groups, Engel elements.The first two authors are supported by the Spanish Government, grant MTM2017-86802-P, partly with FEDER funds. The first author is also supported by the Basque Government,grant IT974-16. The second author is partially supported by the “National Group for Algebraicand Geometric Structures, and their Applications” (GNSAGA - INdAM). The third authoracknowledges the EPSRC (grant number 1652316) for their support. a r X i v : . [ m a t h . G R ] O c t G.A. FERN ´ANDEZ-ALCOBER, M. NOCE, AND G.M. TRACEY
Engel conditions in group theory have to do with the triviality of these iteratedleft normed commutators. If [ g, n x ] = 1 for some n ∈ N , we say that x is Engelon g , and the smallest such n is the Engel degree of x on g . If x is Engel onall elements g ∈ G , we say that x is a left Engel element of G . Observe thatthe Engel degree of x can vary as g runs over G , and in principle could beunbounded. If there is a bound for the Engel degrees of x , i.e. if there exists n ∈ N such that [ g, n x ] = 1 for all g ∈ G , we say that x is a bounded left Engelelement of G . We denote by L( G ) and L( G ) the sets of left Engel elements andbounded left Engel elements of G , respectively. On the other hand, if g ∈ G is such that every x ∈ G is Engel on g , we say that g is a right Engel element of G . Bounded right Engel elements are defined in an obvious way. We writeR( G ) and R( G ) for the sets of right Engel elements and bounded right Engelelements of G . Observe that R( G ) ⊆ L( G ) − [17, 12.3.1], and that obviouslyL( G ) ⊆ L( G ) and R( G ) ⊆ R( G ). In particular, if L ( G ) = 1 then all four Engelsets are trivial.We say that G is an Engel group if L ( G ) = G (or equivalently R ( G ) = G ).On the other hand, if the identity [ g, x, n . . ., x ] = 1 holds for all g, x ∈ G , i.e.if every x ∈ G is a bounded left Engel element with a common bound for all g ∈ G (or equivalently with the right Engel condition), then G is said to be an n -Engel group .In every group G , the sets L( G ), L( G ), R( G ), and R( G ) contain some dis-tinguished subgroups, namely the Hirsch-Plotkin radical, the Baer radical, thehypercenter and the ω -center, respectively. In his book A Course in the Theoryof Groups , D.J.S. Robinson considers it one of the major goals of Engel theoryto find conditions which will guarantee that these four sets of Engel elementscoincide with the corresponding subgroups [17, Section 12.3]. For example, Baerproved that this is the case if G satisfies the maximal condition (see [3, Satz L (cid:48) ]or [17, 12.3.7]); in particular, L( G ) coincides with the Fitting subgroup if G isfinite. However, these equalities do not hold in general. It is then natural to askwhether L( G ), L( G ), R( G ), and R( G ) are always subgroups of G , and it wasnot until recently that the first counterexamples were found. It is here wherebranch groups come into play in Engel theory.Let G be the first Grigorchuk group. This is a branch group acting on thebinary tree, introduced by Grigorchuk [9] in 1980. In 2006 Bludov announced[6] that the wreath product G (cid:111) D , with the natural action of D on 4 points,can be generated by Engel elements but is not an Engel group. In particular,L( G ) is not a subgroup. This example was never published, but ten years later,Bartholdi [4] showed that L( G ) = { x ∈ G | x = 1 } and, as a consequence, that L( G ) is not a subgroup. To date, it is still an openquestion whether L( G ), R( G ) and R( G ) are always subgroups.The other major question in Engel theory is whether Engel groups are locallynilpotent. The answer is negative in general, the main example being the so-called Golod-Shafarevich groups. These groups also provide a negative answerto the General Burnside Problem (GBP), which can be equivalently formulatedas the question of whether periodic groups are locally finite. This resemblancebetween Engel and Burnside problems, together with the fact that many groups NGEL ELEMENTS IN WEAKLY BRANCH GROUPS 3 answering GBP in the negative are branch groups, and Bartholdi’s result on leftEngel elements of the Grigorchuk group, makes it natural to study the behaviourof the Engel sets L( G ), L( G ), R( G ) and R( G ) in (weakly) branch groups. Thisis the specific goal that we are addressing in this paper.Before proceeding to state our main theorems, let us mention some results inthe literature regarding Engel elements in groups of automorphisms of spher-ically homogeneous rooted trees. In the aforementioned paper, Bartholdi alsoproved that, if G is the Gupta-Sidki 3-group, then L( G ) = 1. On the other hand,in [7], Garreta and the first two authors proved that again L( G ) = 1 if G is anyfractal subgroup of a Sylow pro- p subgroup of the group of automorphisms ofthe p -adic tree satisfying the condition | G (cid:48) : st G (1) (cid:48) | = ∞ , and in particular if G is non-abelian and has torsion-free abelianization. Also, Tortora and the secondauthor showed in [16] that L( G ) = R( G ) = 1 for the Grigorchuk group G . Aswe next see, the situation in these classes of groups generalises to a great extentto weakly branch groups, which have a tendency to have trivial Engel sets.Our first main result reads as follows. Theorem A.
Let G be a weakly branch group. Then the following hold: (i) L( G ) = 1 . (ii) If the set of finite order elements of L( G ) is non-trivial then it is a p -setfor some prime p , and the rigid stabilizer rst G ( n ) is a p -group for some n ≥ . Thus even if weakly branch groups provide examples in which L( G ) is not asubgroup, they cannot be used to obtain similar examples for L( G ). This resultcan be interpreted in a similar vein to the fact that weakly branch groups, beingresidually finite, cannot provide examples of finitely generated infinite groupsof finite exponent: the “problem” in both cases is boundedness. On the otherhand, part (ii) of Theorem A raises the following question. Question 1.
Can a weakly branch group G contain left Engel elements of infiniteorder? If the answer is negative, then L( G ) consists entirely of p -elements forsome prime p . If instead of weakly branch the group is actually branch, then we have thefollowing stronger version of Theorem A.
Theorem B.
Let G be a branch group. If L( G ) (cid:54) = 1 then G is periodic andthere exists a prime p such that: (i) L( G ) consists of p -elements. (ii) G is virtually a p -group. Compare the results in Theorem B with the situation in the Grigorchuk group.In that case, L( G ) consists of all elements of order 2 in G , and G is a 2-group.On the other hand, the prime p can be arbitrary in (i) and (ii): if F p is thegroup of p -finitary automorphisms of a p -adic tree then it is easy to see thatL( F p ) = F p , and this is a p -group. Observe however that, contrary to G , thegroup F p is not finitely generated. Question 2.
Are there any finitely generated (weakly) branch groups for whichthe set L( G ) is non-trivial and consists of p -elements for an odd prime p ? G.A. FERN ´ANDEZ-ALCOBER, M. NOCE, AND G.M. TRACEY
In the following theorem we consider right Engel elements in weakly branchgroups under a relatively mild condition.
Theorem C.
Let G be a weakly branch group. If the rigid stabilizer rst G ( n ) isnot an Engel group for any n ∈ N , then R( G ) = 1 . Question 3. Is R( G ) = 1 for every finitely generated (weakly) branch group?By Theorem C, this seems closely linked to this other question: can a finitelygenerated (weakly) branch group be Engel? Again, without finite generation, the group F p shows that the answer is neg-ative in both cases. Regarding the last question, observe that weakly branchgroups cannot satisfy a law (see [1, Corollary 1.4] or [14]) and so cannot be n -Engel for a fixed n . Thus we are asking whether finite generation makes itimpossible for them to be Engel as well.As an application of Theorems A, B, and C, we get the following corollary,which provides information about Engel elements in some specific families ofweakly branch groups. The definition of these families is given either in Section2 or right before the proof of the corresponding result. Corollary D.
Let T be a spherically homogeneous rooted tree. Then the fol-lowing hold: (i) If G is an infinitely iterated wreath product of finite transitive permu-tation groups of degree at least , then L( G ) = 1 . This applies inparticular to the whole group of automorphisms of T , and also to itsSylow pro- p subgroups if T is a p -adic tree, where p is a prime. (ii) If F is the group of finitary automorphisms of T , and there are infinitelymany levels in which the number of descendants is greater than , then L( F ) = 1 . If T is a p -adic tree and F p is the group of p -finitaryautomorphisms of T , then L( F p ) = 1 . (iii) If H is the Hanoi Tower group then L( H ) = 1 . (iv) If G is a multi-GGS groups then R( G ) = 1 , and if G is furthermorenon-periodic then L( G ) = 1 . We conclude this introduction by indicating how the paper is organised. InSection 2 we first give some generalities about groups of automorphisms of spher-ically homogeneous rooted trees, with special emphasis on weakly branch groups.Then we provide several results regarding orbits of such automorphisms that willbe essential later on. Our approach to the study of Engel elements in weaklybranch groups is through the reduction to wreath products. Section 3 is de-voted to a careful analysis of the scenarios that will arise when we apply thiskind of reduction. Finally, in Section 4 we prove Theorems A and B, regardingleft Engel elements in weakly branch groups, and then in Section 5 we obtainTheorem C about right Engel elements. The proof of the applications given inCorollary D is split between these two sections.
Notation and terminology. If f : X → Y and g : Y → Z are two maps, we write f g for their composition instead of g ◦ f . As usual, S n stands for the symmetricgroup on n letters. We denote the direct product of groups G , . . . , G n by (cid:81) ni =1 G i . Given a group G , an element g ∈ G is a p -element , where p is a prime, NGEL ELEMENTS IN WEAKLY BRANCH GROUPS 5 if its order is a power of p , and a subset X of G is a p -set if every element of X is a p -element. Also, if G is finite, F ( G ) denotes the Fitting subgroup of G .2. Automorphisms of spherically homogeneous rooted trees
In this section, we first give some notation and general facts about groups ofautomorphisms of a spherically homogeneous rooted tree, and more specifically,about (weakly) branch groups. For further information on the topic one cansee, for example, [5] or [10]. Then we provide some results regarding orbits ofautomorphisms of such trees that will be needed in the following sections.Let d = { d n } ∞ n =1 be an infinite sequence of integers greater than 1, andlet d = d . We write T d to denote the spherically homogeneous rooted tree corresponding to d . This is a rooted tree where all vertices at level n (i.e. atdistance n from the root) have the same number d n +1 of immediate descendants.If d takes the constant value d , we write T d for T d , and we call it the d -adictree . In order to ease notation, and unless it is strictly necessary to make thesequence d explicit, all throughout the paper we will simply write T to denotean arbitrary spherically homogeneous rooted tree. Also we will write V ( T ) forthe set of vertices of T and, for every n ∈ N , we let L n be the set of all verticeson the n th level of T . d d d d d ∅ ... ...... ... ... ...... ... . . .. . . . . . . . .. . .. . .. . . . . . Figure 1.
A spherically homogeneous rooted treeLet Aut T be the group of automorphisms of T (i.e. bijective maps from V ( T ) to itself that preserve the root and incidence) under the operation ofcomposition. Every f ∈ Aut T can be described by providing, at every vertex v of the tree, the permutation f ( v ) that indicates how f sends the descendantsof v onto the descendants of f ( v ). This permutation is called the label of f at v , and if v lies at level n then f ( v ) ∈ S d n +1 . The collection of all labels of f constitutes the portrait of f , and there is a one-to-one correspondence betweenautomorphisms of T and portraits. An automorphism of T is called finitary ifit has finitely many non-trivial labels in its portrait. Finitary automorphismsform a locally finite subgroup F of Aut T . If T is a p -adic tree for a prime p and we fix a p -cycle σ in S p , the group F p of finitary automorphisms whoselabels are all powers of σ constitute a subgroup of F . We call this the groupof p -finitary automorphisms of T . (We give no reference to σ , since different G.A. FERN ´ANDEZ-ALCOBER, M. NOCE, AND G.M. TRACEY choices of the p -cycle give rise to isomorphic groups.) Observe that F p is locallya finite p -group.We write st( v ) for the stabilizer of v ∈ V ( T ) in Aut T and st( n ) for thepointwise stabilizer of L n , i.e.st( n ) = ∩ v ∈L n st( v ) . The latter is a normal subgroup of finite index of Aut T . The factor groupAut T / st( n ) is naturally isomorphic to the automorphism group of the finitetree consisting of all levels of T up to (and including) the n th level. Then Aut T is isomorphic to the inverse limit of these finite groups, and is so a profinitegroup. Also, we haveAut T ∼ = · · · ( S d n (cid:111) ( · · · ( S d (cid:111) ( S d (cid:111) S d )) · · · )) · · · , where the iterated wreath product is permutational at every step. If we considerthe p -adic tree T p , where p is a prime, and we consider a fixed p -cycle σ ∈ S p ,then the set Γ p of all automorphisms of T with labels in (cid:104) σ (cid:105) is a Sylow pro- p subgroup of Aut T p . We say that Γ p is a standard Sylow pro- p subgroup ofAut T p . Observe thatΓ p ∼ = · · · ( C p (cid:111) ( · · · ( C p (cid:111) ( C p (cid:111) C p )) · · · )) · · · . Let T v be the subtree hanging from the vertex v of the tree. We have T u ∼ = T v for any two vertices u , v on the same level, and we denote by T (cid:104) n (cid:105) any treeisomorphic to a subtree with root in L n . If s denotes the shift operator thaterases the first term of a sequence, then T (cid:104) n (cid:105) is isomorphic to the sphericallyhomogeneous tree defined by the sequence s n ( d ).Every f ∈ Aut T naturally induces a bijection between T v and T f ( v ) which,under the identification of these trees with T (cid:104) n (cid:105) , defines an automorphism f v of T (cid:104) n (cid:105) . This is called the section of f at v . Sections satisfy the following rules, forall f, g ∈ Aut T and v ∈ V ( T ): ( f g ) v = f v g f ( v ) and ( f g ) g ( v ) = ( g v ) − f v g f ( v ) . As a consequence, if f ∈ st( v ), we get(2.1) ( f g ) g ( v ) = ( f v ) g v . If f fixes the vertex v , then the section f v is nothing but the restriction of f to T v . The assignment f (cid:55)→ f v induces a homomorphism ψ v : st( v ) → Aut T (cid:104) n (cid:105) ,and the map ψ n : st( n ) −→ Aut T (cid:104) n (cid:105) × d ...d n · · · × Aut T (cid:104) n (cid:105) f (cid:55)−→ ( f v ) v ∈L n . is an isomorphism. If n = 1 we simply write ψ for ψ . In the case of a d -adictree, we get st( n ) ∼ = Aut T × d n · · · × Aut T . NGEL ELEMENTS IN WEAKLY BRANCH GROUPS 7
Observe also that Aut T splits over st( n ) for every n ≥
1. One can take as acomplement the subgroup H n = { f ∈ Aut
T | f v = 1 for all v ∈ L n } = { f ∈ Aut
T | f ( v ) = 1 for all v ∈ ∪ i ≥ n L i } . The automorphisms in H are called rooted automorphisms of T . They act on T by permuting rigidly the subtrees hanging from the root according to somepermutation of S d . Every automorphism f ∈ Aut T can be uniquely writtenin the form gh , where g ∈ st(1) and h is rooted. If ψ ( g ) = ( g , . . . , g d ) and h corresponds to a permutation σ ∈ S d , we use the following shorthand notationto denote f : f = ( g , . . . , g d ) σ. Now let G be a subgroup of Aut T . We set st G ( n ) = st( n ) ∩ G for all n ≥ v is a vertex of T , the rigid stabilizer of v in G is defined as follows:rst G ( v ) = { g ∈ G | g ( u ) = u for all u lying outside T v } . If V is a set of vertices, all lying on the same level of T , we setrst G ( V ) = (cid:104) rst G ( v ) | v ∈ V (cid:105) , the rigid stabilizer of V in G . It turns out thatrst G ( V ) = (cid:89) v ∈ V rst G ( v ) . We write rst G ( n ) for the rigid stabilizer of L n , and call it the n th rigid stabilizer of G . It is the direct product of the rigid stabilizers of all vertices of L n , andit is the largest “geometrical” direct product inside st G ( n ), in the sense that asubgroup H of st G ( n ) satisfies ψ n ( H ) = (cid:89) v ∈L n H v with H v ≤ Aut T v if and only if H ≤ rst G ( n ). Obviously, if G is the wholeof Aut T then the n th rigid stabilizer coincides with the n th level stabilizer.However, this is not usually the case for arbitrary subgroups of Aut T .By (2.1), we have(2.2) rst G ( v ) g = rst G ( g ( v ))for every v ∈ V ( T ) and g ∈ G . Thus if G is spherically transitive , i.e. if G acts transitively on every L n , then each level rigid stabilizer rst G ( n ) is a directproduct of isomorphic subgroups for all n ∈ N . We are now ready to introducethe class of groups that are the object of our study. Definition 2.1.
Let G be a spherically transitive subgroup of Aut T . Then:(a) If | G : rst G ( n ) | < ∞ for all n ∈ N , we say that G is a branch group .(b) If rst G ( n ) (cid:54) = 1 for all n ∈ N , we say that G is a weakly branch group .Notice that all rigid level stabilizers in a weakly branch group are infinite.Also, since spherically transitive groups are infinite, branch groups are obviouslyweakly branch.After this quick introduction to groups of automorphisms of a sphericallyhomogeneous rooted tree, we start developing the tools that we will use in the G.A. FERN ´ANDEZ-ALCOBER, M. NOCE, AND G.M. TRACEY proof of Theorems A, B, and C. A key ingredient in our approach to Engelproblems in weakly branch groups is the reduction of the action of an automor-phism f from the whole tree to one or several “reduced trees” determined bysome special orbits of f on V ( T ). For this reason, we start by describing someproperties of orbits of automorphisms of T . Definition 2.2. If f ∈ Aut T and v ∈ V ( T ), the f -orbit of v is the orbit of v under the action of (cid:104) f (cid:105) on V ( T ), i.e. the set { f i ( v ) | i ∈ Z } . The f -orbit is trivial if it consists of only one vertex, that is, if f ( v ) = v .In the statement of the following lemma, we consider the least common mul-tiple of an unbounded family of positive integers to be infinity. Lemma 2.3.
Let f ∈ Aut T and, for every vertex v ∈ V ( T ) , let O v be the f -orbit of v . Then the following hold: (i) If w is a descendant of v , then |O v | divides |O w | . (ii) | f | = lcm( |O v | | v ∈ V ( T )) . (iii) If | f | is finite then there exists a finite subset V of V ( T ) satisfying that | f | = lcm( |O v | | v ∈ V ) and that, whenever w is a descendant of avertex v ∈ V , we have |O w | = |O v | . Also if f is non-trivial then all theorbits O v with v ∈ V are non-trivial. Furthermore, V can be chosen tolie in L n for some n .Proof. (i) This is obvious by the orbit-stabilizer theorem, since st( w ) ⊆ st( v ).(ii) Set H = (cid:104) f (cid:105) . Then |O v | = | H/ st H ( v ) | for all v ∈ V ( T ). The naturalmap ϕ from H to the cartesian product of finite groups (cid:81) v ∈ V ( T ) H/ st H ( v ) isinjective, since the intersection of all vertex stabilizers is trivial. Consequently | f | = | ϕ ( f ) | = lcm( | f st H ( v ) | | v ∈ V ( T )) = lcm( | H/ st H ( v ) | | v ∈ V ( T )) , which proves the result.(iii) Let L = {|O v | | v ∈ V ( T ) } . If | f | is finite then, by (ii), it can be achievedas the least common multiple of a finite subset of L . Let k be the minimumcardinality of such a subset and let S = { S ⊆ L | | S | = k and lcm( S ) = | f |} . Observe that S is a finite set.We introduce a relation ≤ d in S by letting S ≤ d T if there exists a bijection α : S → T such that s | α ( s ) for all s ∈ S . By (i), this models the situation whenwe pass from the orbits of a set of vertices to the orbits of a set of descendantsof those vertices. We claim that ≤ d is an order relation in S . Obviously, onlyantisymmetry needs to be checked. Assume that α : S → T and β : T → S aresuch that s | α ( s ) and t | β ( t ) for all s ∈ S and t ∈ T . Then s divides β ( α ( s ))and, if they are not equal, we get lcm( S (cid:114) { s } ) = | f | . This is contrary to theminimality condition imposed on k . Thus β ( α ( s )) = s and, since s | α ( s ) and α ( s ) | β ( α ( s )), we obtain that α ( s ) = s for all s ∈ S . We conclude that S = T ,which proves antisymmetry of ≤ d .Now choose S in S that is maximal with respect to the order ≤ d , and let V = { v , . . . , v k } ⊆ V ( T ) be such that S = {|O v | , . . . , |O v k |} . Consider anarbitrary set of vertices W = { w , . . . , w k } , where each w i is a descendant of v i ,and let T = {|O w | , . . . , |O w k |} . Then S ≤ d T and, by the maximality of S , we NGEL ELEMENTS IN WEAKLY BRANCH GROUPS 9 have S = T . This implies that |O w i | = |O v i | for all i = 1 , . . . , k . Observe alsothat the minimality of k implies that, if f is non-trivial, no orbit O v with v ∈ V is of length 1. Hence V satisfies the properties stated in (iii).Finally, observe that also the set W satisfies the required properties. Thusby considering, for a suitable n , a subset of L n consisting of one descendant ofeach vertex in V , we may assume that V ⊆ L n . (cid:3) Vertices and orbits as in part (iii) of the previous lemma will play a fun-damental role in the rest of the paper, and it is convenient to introduce someterminology.
Definition 2.4.
Let f ∈ Aut T and let O be an f -orbit. We say that O is totally splitting if for every descendant w of a vertex v ∈ O , the length of the f -orbit of w is equal to |O| .Equivalently, an f -orbit O is totally splitting when the set of descendants ofthe vertices in O at every level of the tree splits into the maximum possiblenumber of f -orbits. Definition 2.5.
Let f ∈ Aut T be an automorphism of finite order. If V is afinite set of vertices satisfying the conditions in (iii) of Lemma 2.3, all of themlying on the same level of T , we say that V is a fundamental system of vertices for f .Next we give a sufficient condition for two automorphisms of T to generatea wreath product. Lemma 2.6.
Let f ∈ Aut T be an automorphism of finite order m , and assumethat the f -orbit of a vertex v ∈ V ( T ) has length m . Then for every g ∈ rst( v ) ,the subgroup (cid:104) g, f (cid:105) of Aut T is isomorphic to the regular wreath product (cid:104) g (cid:105) (cid:111) (cid:104) f (cid:105) .Proof. Let O be the f -orbit of v . Since |O| = | f | , we have (cid:104) f (cid:105) ∩ st( v ) = 1. As aconsequence, if v lies at level n of the tree, also (cid:104) f (cid:105) ∩ st( n ) = 1 and(2.3) (cid:104) g, f (cid:105) = (cid:104) f (cid:105) (cid:104) g, g f , . . . , g f m − (cid:105) = (cid:104) f (cid:105) (cid:110) (cid:104) g, g f , . . . , g f m − (cid:105) , since g ∈ rst( v ) implies that (cid:104) g, g f , . . . , g f m − (cid:105) ⊆ st( n ).Now set v i = f i ( v ) for all i ∈ Z , so that O = { v , v , . . . , v m − } . Since g ∈ rst G ( v ), from (2.2) we get g f i ∈ rst G ( v i ) for all i = 0 , . . . , m −
1, and then (cid:104) g f i (cid:105) ∩ (cid:104) g, g f , . . . , g f i − (cid:105) ⊆ rst G ( v i ) ∩ rst G ( { v , . . . , v i − } ) = 1 . Also [ g f i , g f j ] = 1 for every i, j ∈ { , . . . , m − } . It follows that (cid:104) g, g f , . . . , g f m − (cid:105) = (cid:104) g (cid:105) × (cid:104) g f (cid:105) × · · · × (cid:104) g f m − (cid:105) , and since g f m = g , we conclude from (2.3) that (cid:104) g, f (cid:105) ∼ = (cid:104) g (cid:105) (cid:111) (cid:104) f (cid:105) . (cid:3) The result in Lemma 2.6 raises the question of whether an automorphism f ∈ Aut T of finite order m must have a regular orbit on V ( T ), i.e. an orbit oflength m . This is clearly the case if m is a prime power, by (ii) of Lemma 2.3,but it usually fails otherwise. Indeed, one can consider for example a rootedautomorphism corresponding to a permutation whose order is strictly biggerthan the lengths of its disjoint cycles. However, as we see in Lemma 2.9 below,it is always possible to derive a collection of automorphisms f i from f , acting not on T but on some other rooted trees R i obtained from T , and having theproperty that every f i has a regular orbit on V ( R i ). These automorphisms f i will allow us to study Engel conditions regarding f by using Lemma 2.6.As we will see, Lemma 2.9 is essentially a reformulation of (iii) of Lemma 2.3.Before proceeding we need to introduce the concept of reduced tree. Note thatreduced trees are somehow related to the trees obtained by deletion of layersdefined by Grigorchuk and Wilson in [12]. Definition 2.7.
Let V be a subset of vertices of T , all lying on the same level n . We define the reduced tree of T at V , denoted by R ( V ), as the rooted treeconsisting of the subtrees T v for v ∈ V , all connected to a common root. Inother words, the set of vertices of R ( V ) is {∅} ∪ { vw | v ∈ V, w ∈ T s n +1 ( d ) } , where as before s denotes the shift operator on sequences.For example, in the following figure, we consider the rooted automorphism f of the ternary tree T corresponding to the permutation (1 2 3) and we show inred the reduced tree at the orbit of the vertex 13: ... ... ... ... ... ... ... ... ...... ... ... ... ... ... ... ... ...... ... ... ... ... ... ... ... ...... ... ... ... ... ... ... ... ... Figure 2. An f -orbit and its corresponding reduced treeEvery f ∈ Aut T such that f ( V ) = V induces by restriction an automorphism f V ∈ Aut R ( V ). Clearly, the map Φ V : f (cid:55)−→ f V is a homomorphism of groups.The effect of Φ V is to focus on the action of f only on the subtrees T v with v ∈ V , so to speak. We will use reduced trees mainly in the case where V is anorbit of f . Remark 2.8. If v is a vertex of the reduced tree R ( V ) and f ∈ Aut T is suchthat f ( V ) = V , then the f V -orbit of v coincides with the f -orbit of v as a vertexin V ( T ). In particular, if O is a totally splitting f -orbit and we consider theinduced automorphism x = Φ O ( f ) of R ( O ), then (ii) of Lemma 2.3 implies that | x | = |O| . In other words, O is a regular orbit of x in R ( O ).Given a subgroup G of Aut T , we write G V for the image of the setwisestabilizer of V in G under the homomorphism Φ V . In other words, G V = { f V | f ∈ G and f ( V ) = V } . NGEL ELEMENTS IN WEAKLY BRANCH GROUPS 11
Then G V is a subgroup of Aut R ( V ), and for every vertex v ∈ V we haveΦ V (rst G ( v )) ⊆ rst G V ( v ) (the inclusion can be proper, since there can be auto-morphisms in G whose action is trivial on T w for every w (cid:54) = v with w ∈ V , butnon-trivial for some w (cid:54)∈ V ).On the other hand, if f ∈ G stabilizes the set V and x = Φ V ( f ) is the inducedautomorphism of R ( V ), then f ∈ L( G ) or f ∈ L( G ) imply that x ∈ L( H ) or x ∈ L( H ), respectively. In particular, by choosing V to be an f -orbit, thiswill allow us to transfer the analysis of a given Engel element in a subgroup ofAut T to a more restricted situation where, for example, the Engel element actstransitively on the first level of the tree.Actually the most convenient strategy is to reduce the tree to non-trivialtotally splitting f -orbits, since the induced automorphisms will then have regularorbits. More precisely, we will rely on the following lemma, which is basically arephrasing of part of Lemma 2.3 in the language of reduced trees. Lemma 2.9.
Let f ∈ Aut T be an automorphism of finite order m > and let { v , . . . , v k } be a fundamental system of vertices for f . For every i = 1 , . . . , k ,let O i be the f -orbit of v i , set R i = R ( O i ) , and let f i be the automorphism of R i induced by f . Then the following hold: (i) lcm( |O | , . . . , |O k | ) = m . (ii) O i is a non-trivial totally splitting f -orbit for every i = 1 , . . . , k . (iii) | f i | = |O i | for every i = 1 , . . . , k .Proof. The first two items follow from (iii) of Lemma 2.3, and (iii) from Re-mark 2.8. (cid:3) Some properties of Engel elements in wreath products
In this section we prove several results regarding Engel elements in wreathproducts. These will provide the basis for the proof of the main theorems inthis paper, which will be addressed in Sections 4 and 5.We start by studying left Engel elements lying outside the base group of aregular wreath product of two cyclic groups. To this purpose, we rely on thepaper [15] by Liebeck.
Lemma 3.1.
Let X = (cid:104) x (cid:105) and Y = (cid:104) y (cid:105) be two non-trivial cyclic groups, where X is finite, and let W = Y (cid:111) X be the corresponding regular wreath product. If x ∈ L( W ) then X and Y are finite p -groups for some prime p . Furthermore,the Engel degree of x on g = ( y, , . . . , is equal to | x | + 1 p (log p | y | − p − | x | . Proof.
Let m be the order of x , and let p be an arbitrary prime divisor of m .Also, write d for the Engel degree of x on g = ( y, , . . . , Y is finite. Then W is finite and, by Baer’s theoremmentioned in the introduction, x lies in the Fitting subgroup F ( W ). We claimthat Y is then a p -group. To this purpose, assume that | Y | is divisible by aprime q (cid:54) = p , and let Z = (cid:104) z (cid:105) (cid:54) = 1 be the subgroup of Y of order q . Consider thedirect product Z X inside the base group of W . Since Z X is abelian and normalin W , it lies in F ( W ). Now Z X is a q -group and x p = x m/p is a p -element, and both lie in the nilpotent group F ( W ). It follows that x p centralizes Z X , whichis clearly a contradiction, since x p does not commute with ( z, , . . . , m , italso follows that X is a p -group. Observe that, since both X and Y are finite p -groups, the proof of Theorem 5.1 of [15] yields that(3.1) d = m + 1 p (log p | y | − p − m in this case.Now it is easy to see that Y cannot be infinite. For a contradiction, supposethat Y is infinite and consider a prime q different from p . Then the wreathproduct W q = ( Y /Y q ) (cid:111) X can be seen as a factor group of W , and so x is a leftEngel element in W q . Since | Y /Y q | = q and p divides | X | , we get a contradictionwith the previous paragraph. (cid:3) Now we digress from Engel elements for a moment, but still working withwreath products of cyclic groups, in order to prove that rigid stabilizers of weaklybranch groups are not only infinite, but have infinite exponent (Proposition 3.3below).
Lemma 3.2.
Let X = (cid:104) x (cid:105) and Y = (cid:104) y (cid:105) be two finite cyclic groups, where Y isnon-trivial, and let W = Y (cid:111) X be the corresponding regular wreath product. If g = ( y, , . . . , then | xg | > | x | .Proof. Set m = | x | and let n ∈ { , . . . , m } be arbitrary. Then( xg ) n = x n g n g ( n ) . . . g ( nn − ) n − g n , where g = g and g i = [ g, x, i − . . ., x ] for every i = 2 , . . . , n . Now observe that each g i is of the form g i = ( ∗ , . . . , ∗ , y, , . . . , , where we use ∗ to denote unspecified powers of y , and y occupies the i th position.It follows that ( xg ) n = x n ( ∗ , . . . , ∗ , y, , . . . , , where y appears at the n th position. In particular, ( xg ) n (cid:54) = 1 for 1 ≤ n ≤ m ,and consequently | xg | > m , as desired. (cid:3) Proposition 3.3.
Let G be a weakly branch group. Then the exponent of rst G ( n ) is infinite for every n ∈ N .Proof. By way of contradiction, assume that rst G ( n ) has finite exponent. Thusrst G ( n ) is periodic and there is a bound for the orders of its elements. For every k ≥ n , let π k be the (finite) set of prime divisors of the orders of the elementsof rst G ( k ). Then { π k } k ≥ n is a decreasing sequence of non-empty finite sets andconsequently their intersection is also non-empty. Let p be a prime in ∩ k ≥ n π k .Consider a p -element f ∈ rst G ( n ) of maximum order, say m . Since the orderof f is the least common multiple of the orders of the components of ψ n ( f ), wemay assume without loss of generality that f ∈ rst G ( u ) for some vertex u ofthe n th level. By (ii) of Lemma 2.3, there is a vertex v in the tree T such thatthe f -orbit of v has length m . Of course, v must be a descendant of u . Nowthe choice of p allows us to consider a non-trivial p -element g in rst G ( v ). Set NGEL ELEMENTS IN WEAKLY BRANCH GROUPS 13 H = (cid:104) g, f (cid:105) . By Lemma 2.6, we have H ∼ = (cid:104) g (cid:105) (cid:111) (cid:104) f (cid:105) . In particular, H is a finite p -group. On the other hand, by Lemma 3.2, H contains an element of ordergreater than m . This contradicts the choice of m , since H ⊆ rst G ( n ). (cid:3) Now we continue with our analysis of Engel elements in some wreath products.Before proceeding, we introduce some further notation. If G is a group and S ⊆ G , we write L G ( S ) to denote the set of all x ∈ G that are left Engelelements on every element of S , that is, such that for all s ∈ S there exists n = n ( s, x ) such that [ s, n x ] = 1. We define the set L G ( S ) in the obvious way,and if x ∈ L G ( S ) then the Engel degree of x on S is the maximum of the Engeldegrees of x on the elements of S . Lemma 3.4.
Let W = Y (cid:111) X be a regular wreath product of two non-trivialgroups, where X is finite cyclic of order n , and let π : W → X be the naturalprojection. Assume that D = D × · · · × D n (cid:54) = 1 is a subgroup of the base groupof W , and that w ∈ W is such that π ( w ) is a generator of X . Then the followinghold: (i) If w ∈ L W ( D ) has Engel degree d on D then d ≥ n . (ii) If w ∈ L W ( D ) then C D ( w ) is periodic.Proof. Write w = ( y , . . . , y n ) x , where y i ∈ Y and x generates X . We mayassume that x permutes the components of the base group according to thecycle (1 2 . . . n ).(i) Without loss of generality, we may assume that D (cid:54) = 1. Choose a non-trivial element g = ( y, , . . . , ∈ D and let 1 ≤ i ≤ n −
1. One can easily checkby induction on i that[ g, i w ] = ( y ( − i , . . . , y y ...y i , , . . . , , where the last non-trivial component is in position i +1. It follows that [ g, n − w ] (cid:54) =1 and d ≥ n .(ii) By contradiction, assume that h = ( z , . . . , z n ) ∈ C D ( w ) is of infiniteorder. For notational convenience, set z = z n and y = y n . Then from thecondition h = h w we get z i = z y i − i − for all i = 1 , . . . , n . Hence all components of h are conjugate and they are all of infinite order.Now let g = ( z , , . . . , ∈ D . For every k ≥
0, let us write [ g, k w ] =( z k, , . . . , z k,n ) and, as before, set z k, = z k,n . We claim that the following holdfor every k ≥ z k,i ∈ (cid:104) z i (cid:105) for every i = 1 , . . . , n .(b) If we write z k,i = z m k,i i , then there exists i ∈ { , . . . , n } such that m k,i (cid:54) = m k,i − .We argue by induction on k . The result is obvious for k = 0, so assume k ≥ k . Since [ g, k w ] = [ g, k − w ] − [ g, k − w ] w ,it follows that z k,i = z − k − ,i z y i − k − ,i − = z − m k − ,i i ( z y i − i − ) m k − ,i − = z m k − ,i − − m k − ,i i for all i = 1 , . . . , n . This proves (a) and, if (b) does not hold, then m k − , − m k − , = m k − , − m k − , = · · · = m k − ,n − − m k − ,n = m k − ,n − m k − , . Now the sum of the n − n − m k − , − m k − ,n = ( n − m k − ,n − m k − , ) . From this, it readily follows that m k − , = m k − , = m k − , = · · · = m k − ,n , which is contrary to the induction hypothesis.Finally, observe that (b) above implies that m k,i and m k,i − cannot bothbe zero. Since z i and z i − are of infinite order, we conclude that [ g, k w ] (cid:54) = 1for all k ≥ w (cid:54)∈ L W ( D ). This contradiction completes theproof. (cid:3) Left Engel elements in weakly branch groups
At this point, we can start combining all the machinery developed in Sections2 and 3 in order to prove the main results of this paper. In this section we con-sider left Engel elements. The following is an expanded version of Theorem A.
Theorem 4.1.
Let G be a subgroup of Aut T in which all rigid vertex stabilizersare non-trivial. Then: (i) If f is a non-trivial left Engel element of finite order, and O is a non-trivial totally splitting f -orbit, then for some prime number p the lengthof O is a p -power and rst G ( O ) is a p -subgroup.If G is furthermore weakly branch, then: (ii) If the set of finite order elements of L( G ) is non-trivial then it is a p -setfor some prime p , and rst G ( n ) is a p -group for some n ≥ . (iii) L( G ) = 1 .Proof. (i) Denote the reduced tree R ( O ) by R , and set x = Φ O ( f ) and H = G O .We observe that | x | = |O| by Remark 2.8. Consider now a vertex v in O and anarbitrary element g ∈ rst G ( v ), and set y = ψ v ( g ) (here v is considered as a vertexin T ). Then h = Φ O ( g ) lies in rst H ( v ) and ψ v ( h ) = y (here v is considered asa vertex in R ). By Lemma 2.6, we have (cid:104) h, x (cid:105) ∼ = (cid:104) h (cid:105) (cid:111) (cid:104) x (cid:105) . Since x ∈ L( H ),Lemma 3.1 implies that both | y | and | x | are p -powers for some prime p . Thus | g | and |O| are p -powers. Since g ∈ rst G ( v ) was arbitrary and f acts transitivelyon O , we conclude that rst G ( O ) is a p -group.(ii) Let again f ∈ L( G ) be a non-trivial element of finite order. By applyingLemma 2.9 to f , we obtain non-trivial totally splitting f -orbits O , . . . , O k , alllying on the same level n of the tree, such that | f | = lcm( |O | , . . . , |O k | ). Letus fix i ∈ { , . . . , k } . By (i), there exists a prime p (in principle, depending on i ) such that |O i | is a p -power and rst G ( O i ) is a p -group. Since G acts now leveltransitively on T , all rigid vertex stabilizers are isomorphic by (2.2). It followsthat p is the same for all i and consequently rst G ( n ) is a p -group. Also thelength of all orbits O , . . . , O k is a power of p and f is a p -element.(iii) By contradiction, assume that f ∈ L( G ), f (cid:54) = 1. Let d be the Engeldegree of f . NGEL ELEMENTS IN WEAKLY BRANCH GROUPS 15
Assume first that f is of finite order. Let O be a non-trivial totally splitting f -orbit. Define x and y as in the proof of (i), and recall that these are p -elements.By Lemma 3.1, d ≥ | x | + 1 p (log p | y | − p − | x | . On the other hand, since the exponent of rst G ( n ) is not finite by Proposition 3.3,the order of y is unbounded. This is a contradiction.Assume now that the order of f is infinite. By Lemma 2.3, there exists an f -orbit O of length (cid:96) > d . Let once again R be the reduced tree R ( O ), and set h = Φ O ( f ) and H = G O . Then h ∈ S (cid:104) x (cid:105) , where S is the first level stabilizer inAut R (i.e. the stabilizer of O ) and x is a rooted automorphism correspondingto a cycle of length (cid:96) . Observe that S (cid:104) x (cid:105) is isomorphic to a regular wreathproduct W = Y (cid:111) X , where Y is the stabilizer in R of a vertex in O and X = (cid:104) x (cid:105) is cyclic of order (cid:96) . Under this isomorphism, h corresponds to an element w with π ( w ) = x . Also h lies in L H ( D ) with Engel degree at most d , where D = Φ O (rst G ( O )) corresponds to a non-trivial direct product inside the basegroup of W . Now, by applying (i) of Lemma 3.4, we get d ≥ (cid:96) , which is acontradiction. This completes the proof of (iii). (cid:3) Now we proceed to prove Theorem B.
Theorem 4.2.
Let G be a branch group. If L( G ) (cid:54) = 1 then G is periodic andthere exists a prime p such that: (i) L( G ) consists of p -elements. (ii) G is virtually a p -group.Proof. It suffices to show that L( G ) does not contain any elements of infiniteorder. Indeed, since L( G ) (cid:54) = 1, the theorem then follows immediately from (ii)of Theorem 4.1, by taking into account that | G : rst G ( n ) | is always finite if G isa branch group.Let us assume then that f ∈ L( G ) is of infinite order. Consider an f -orbit O in V ( T ) of length (cid:96) ≥
2, and let n be the level of T containing O . Set R = R ( O ), h = Φ O ( f ) and H = G O . Then for every vertex v (cid:54) = ∅ of R wehave Φ O (rst G ( v )) ⊆ rst H ( v ), and consequently all rigid vertex stabilizers of H are non-trivial. Also h ∈ L( H ).If h has finite order, then by (i) of Theorem 4.1, the rigid stabilizer in H ofsome vertex v (cid:54) = ∅ of R is periodic. Consequently rst G ( v ) is periodic, and bylevel transitivity of G , also rst G ( n ) is periodic. Since | G : rst G ( n ) | is finite, itfollows that G itself is periodic, which is a contradiction.Assume now that the order of h is infinite. As in the proof of (iii) of Theo-rem 4.1, h lies in S (cid:104) x (cid:105) , where S is the first level stabilizer of Aut R , and x isa rooted automorphism corresponding to a cycle of length (cid:96) . We can identify S (cid:104) x (cid:105) with the regular wreath product W = Y (cid:111) X , where X = (cid:104) x (cid:105) is cyclic oforder (cid:96) and h maps onto x . Then h ∈ L W ( D ), where D = Φ O (rst G ( O )) = Φ O (rst G ( n ))corresponds to a non-trivial direct product inside the base group of W . By (ii) ofLemma 3.4, C D ( h ) is periodic. However, since G is branch we have f k ∈ rst G ( n )for some k ≥ h k = Φ O ( f k ) ∈ D . It follows that h k ∈ C D ( h ) is anelement of infinite order, which is a contradiction. (cid:3) Now we can apply Theorems A and B to some distinguished subgroups ofAut T , and obtain the part of Corollary D regarding left Engel elements. Beforeproceeding, we will introduce some of the groups that appear in the followingresult.First of all, the Hanoi Tower group H is the subgroup of Aut T generated bythe three automorphisms a , b and c given by the following recursive formulas: a = (1 , , a )(1 2) ,b = (1 , b, ,c = ( c, , . This group models the popular Hanoi Tower puzzle on 3 pegs.On the other hand, given an odd prime p and a non-trivial subspace E of F p − p , we define the multi-GGS group (GGS standing for Grigorchuk, Gupta,and Sidki) G E as the following subgroup of Aut T p . The group G E is generatedby the rooted automorphism a of order p corresponding to the p -cycle (1 2 . . . p ),and by the elementary abelian p -subgroup B consisting of all automorphisms b e , with e = ( e , . . . , e p − ) ∈ E , defined recursively via(4.1) b e = ( a e , . . . , a e p − , b e ) . If dim E = 1 then G E is simply called a GGS group. Multi-GGS groups areusually presented by giving a basis ( e , . . . , e r ) of E and defining b i ∈ B from e i as in (4.1) for each i = 1 , . . . , r , so that G E = (cid:104) a, b , . . . , b r (cid:105) . We refer thereader to the paper [2] by Alexoudas, Klopsch, and Thillaisundaram for generalfacts about multi-GGS groups. Multi-GGS groups are infinite and provide awealth of examples giving a negative answer to the General Burnside Problem.For instance, the famous Gupta-Sidki p -group is the GGS group (cid:104) a, b (cid:105) with b corresponding to the vector e = (1 , − , , . . . , E is contained in the hyperplane of F p − p given by theequation e + · · · + e p − = 0 [2, Theorem 3.2]. On the other hand, multi-GGSgroups are known to be branch unless E = (cid:104) (1 , . . . , (cid:105) consists of constantvectors, in which case it is weakly branch [2, Proposition 3.7]. Corollary 4.3.
In all the following groups, the only left Engel element is theidentity: (i)
Every infinitely iterated wreath product of finite transitive permutationgroups of degree at least . In particular, Aut T and Γ p , for p a prime. (ii) The group F of all finitary automorphisms of T , provided that the se-quence d defining T contains infinitely many terms greater than . (iii) All non-periodic multi-GGS groups G E , i.e. those with at least one vec-tor e ∈ E having non-zero sum in F p . (iv) The Hanoi Tower group H .Proof. (i) For every n ∈ N , let K n be a finite transitive permutation group ofdegree d n ≥
2, and let W be the iterated wreath product of all these groups.Let T be the spherically homogeneous rooted tree corresponding to the sequence d = { d n } n ∈ N . Then W is isomorphic to the subgroup K of Aut T consisting ofall automorphisms whose labels at level n are elements of K n +1 . Observe that K NGEL ELEMENTS IN WEAKLY BRANCH GROUPS 17 is a branch group, since every K n is transitive and obviously rst K ( n ) = st K ( n )in this case.According to Theorem B, we only need to construct an element of infiniteorder in K to conclude that L( W ) = 1. To this purpose, we choose an infinitesequence { k n } n ∈ N of non-trivial permutations k n ∈ K n , and an infinite sequence { v n } n ∈ N ∪{ } of vertices, where v n ∈ L n and k n ( v n ) (cid:54) = v n . Also, let O n denotethe orbit of v n under (cid:104) k n (cid:105) and set (cid:96) n = |O n | .Now we define f to be the automorphism of T having label k n +1 at vertex v n for all n ∈ N ∪ { } . We claim that the length of the f -orbit of v n is (cid:96) . . . (cid:96) n for all n ∈ N . Since (cid:96) i ≥ i , we conclude that f is of infinite order byusing (ii) of Lemma 2.3.We prove the claim by induction on n . The result is obvious for n = 1, since f behaves as k on the first level of T . Then f (cid:96) fixes all vertices in the orbit O , and a simple calculation shows that on all those vertices the section of f (cid:96) coincides with the section of f at v , let us call it g . Since v n lies at level n − g , by induction the length of the g -orbit of v n is (cid:96) . . . (cid:96) n . From this one canreadily see that the f -orbit of v has length (cid:96) . . . (cid:96) n , as desired.(ii) Obviously, F is spherically transitive and rst F ( n ) = st F ( n ) for all n ∈ N .Thus F is a branch group. In this case, all elements of F are of finite order, butwe still get L( F ) = 1 from Theorem B, because there is no prime p for which F is virtually a p -group. Indeed, assume for a contradiction that N is a normal p -subgroup of F of finite index m . Under this assumption, if H is a q -subgroup of F for a prime q (cid:54) = p , the order of H cannot exceed m . However, as we see in thenext paragraph, the condition on the sequence d implies that F has 2-subgroupsand 3-subgroups of arbitrarily high order, and we get a contradiction.Consider the following subset of N : S = { n ∈ N | d n ≥ } . By hypothesis, S is infinite. For every n ∈ S , let H n be the subgroup of F consisting of all automorphisms with labels lying in (cid:104) (1 2) (cid:105) for all vertices in L n and trivial labels elsewhere. Then the order of H n is 2 d ...d n , which tendsto infinity as n → ∞ . We can define similarly a subgroup J n of order 3 d ...d n for every n ∈ S , by using the 3-cycle (1 2 3). Thus we get 2-subgroups and3-subgroups of F of arbitrarily high order, as desired.(iii) If E = (cid:104) (1 , . . . , (cid:105) then L( G E ) = 1 by [7, Theorem 7]. Otherwise G E isa branch group, and the result follows immediately from Theorem B and fromthe characterisation of periodic multi-GGS groups given above.(iv) The Hanoi Tower group is known to be a branch group [11, Theorem 5.1].Let us see that the element ab = ( b, , a )(1 2 3) is of infinite order. Assume, fora contradiction, that | ab | = k is finite. Observe that k = 3 (cid:96) for some (cid:96) , since ab has order 3 modulo the first level stabilizer. But then( ab ) (cid:96) = (( ba ) (cid:96) , ( ab ) (cid:96) , ( ab ) (cid:96) )implies that ( ab ) (cid:96) = 1, which is a contradiction. (cid:3) Corollary 4.4.
Let p be a prime and let F p be the group of p -finitary automor-phisms of Aut T p . Then the following hold: (i) L( F p ) = F p . (ii) L( F p ) = 1 .Proof. Since F p is locally a finite p -group, (i) is clear. On the other hand, since F p is spherically transitive and rst F p ( n ) = st F p ( n ) for all n , (ii) follows directlyfrom Theorem A. (cid:3) Right Engel elements in weakly branch groups
In this final section, we prove Theorem C, regarding right Engel elements inweakly branch groups, and then we apply it to show that R( G ) = 1 whenever G is a GGS group. Before proceeding, we need a straightforward lemma. Lemma 5.1.
Suppose that T has d vertices in the first level, and consider x, y ∈ Aut T such that: (i) y = az , where a is the rooted automorphism corresponding to the cycle (1 2 . . . d ) and z ∈ st(1) is given by ψ ( z ) = ( z , . . . , z d ) . (ii) x ∈ st(1) is given by ψ ( x ) = ( x , . . . , x d ) .Then, for all k ≥ , we have ψ ([ y, k x ]) = ([( x − d ) z , k − x ] x , . . . , [( x − d − ) z d , k − x d ] x d ) . Proof.
We have ψ ([ y, x ]) = ψ (( x − ) y x ) = ψ (( x − ) a ) ψ ( z ) ψ ( x )= (( x − d ) z x , ( x − ) z x , . . . , ( x − d − ) z d x d ) . Now the result follows immediately by observing that taking subsequent com-mutators with x is performed componentwise. (cid:3) Now we are ready to prove Theorem C.
Theorem 5.2.
Let G be a weakly branch group. If rst G ( n ) is not an Engelgroup for all n ∈ N , then R( G ) = 1 .Proof. Let f ∈ G , f (cid:54) = 1, and assume by way of contradiction that f ∈ R( G ).Choose a non-trivial f -orbit O = { v , . . . , v d } , and assume that f permutescyclically the vertices v i . Let R = R O , H = G O and y = Φ O ( f ) ∈ R( H ).Then we can write y = az , where a is rooted in R corresponding to the cycle(1 2 . . . d ) and z is in the first level stabilizer. Write ψ ( z ) = ( z , . . . , z d ).Let n be the level of T where O lies. Since Φ O (rst G ( n )) ⊆ rst H (1) and rst G ( n )is not Engel by hypothesis, it follows that rst H (1) is not an Engel group. If L is the first component of the direct product ψ (rst H (1)) then L is not an Engelgroup either, and we can choose a, b ∈ L such that [ b, k a ] (cid:54) = 1 for all k ≥
1. Nowconsider r , r ∈ rst H (1) such that ψ ( r ) = ( a, , . . . ,
1) and ψ ( r ) = ( b, , . . . , , and define x = r ( r − ) y − , so that ψ ( x ) = ( a, , . . . , , ( b − ) z − ) . By applying the formula in Lemma 5.1, we get ψ ([ y, k x ]) = ([ b, k − a ] a , ∗ , . . . , ∗ )and consequently [ y, k x ] (cid:54) = 1 for all k ≥
2. This is a contradiction, since y ∈ R( H ) and x ∈ H . (cid:3) NGEL ELEMENTS IN WEAKLY BRANCH GROUPS 19
Theorem C can be applied to show that GGS groups have no non-trivial rightEngel elements. We first need to prove the weaker result that they are not Engelgroups.
Lemma 5.3.
Let G be a GGS group. Then G is not an Engel group.Proof. We show that there is a power of b that is not a left Engel element of G . Let e be the defining vector of b . Consider any index i ∈ { , . . . , p − } suchthat e p − i (cid:54) = 0 in F p , and choose λ ∈ F × p such that λe p − i = − i . Then we have(5.1) ψ (( b − λ ) a i ) = ( ∗ , . . . , ∗ , a i ) , where we use ∗ to denote unspecified elements of G .Since ( b − λ ) a i = [ a i , b λ ] b − λ , it follows that, for every k ≥ b − λ ) a i , b λ , k − . . . , b λ ] = [[ a i , b λ ] b − λ , b λ , k − . . . , b λ ]= [[ a i , b λ , b λ ] b − λ , b λ , k − . . . , b λ ]= [ a i , b λ , k . . ., b λ ] b − λ . By using (5.1), it follows that(5.2) ψ ([ a i , b λ , k . . ., b λ ] b − λ ) = ψ ([( b − λ ) a i , b λ , k − . . . , b λ ]) = ( ∗ , . . . , ∗ , [ a i , b λ , k − . . . , b λ ]) . Now if b λ is a left Engel element of G , choose the minimum k ≥ a i , b λ , k . . ., b λ ] = 1. Since a i and b λ do not commute, we have k ≥ a i , b λ , k − . . . , b λ ] (cid:54) = 1. According to (5.2), this is a contradiction. (cid:3) Corollary 5.4.
Let G be a GGS group. Then R( G ) = 1 .Proof. If the defining vector e is constant, then L( G ) = 1 by Theorem 7 of [7],and consequently also R( G ) = 1. Thus in the remainder we assume that e is notconstant. By Lemmas 3.2 and 3.4 of [8], we know that G is regular branch over K , where K = γ ( G ) if e is symmetric and K = G (cid:48) otherwise. Since rst G ( n )contains a copy of K × p n · · · × K for every n ∈ N , if we prove that K is not Engelthen Theorem C applies to conclude that R( G ) = 1.In order to show that K is not Engel, we are going to find a vertex v of thefirst level of the tree such that ψ v ( K ) = G . Since G is not Engel by Lemma 5.3,it follows that K is not Engel either, as desired.We consider separately the cases when e is symmetric and non-symmetric.Assume first that e is non-symmetric, so that K = G (cid:48) . We have ψ ([ b, a ]) = ( a − e b, a e − e , a e − e , . . . , a e p − − e p − , b − a e p − ) . Since e is not constant, there exists i ∈ { , . . . , p − } such that e i (cid:54) = e i +1 in F p .If v is the vertex i + 1 on the first level of the tree, then ψ v ([ b, a ]) = a e i − e i +1 and ψ v ([ b, a ] a i ) = a − e b. Since the subgroup (cid:104) a e i − e i +1 , a − e b (cid:105) coincides with G , we get the desired equality ψ v ( G (cid:48) ) = G . Now let e be symmetric, i.e. such that e i = e p − i for all i = 1 , . . . , p −
1. Since e is not constant, this implies that p ≥
5. We have ψ ([ b, a, a ]) = ψ ([ b, a ] − ) ψ ([ b, a ] a )= ( b − a e b − a e p − , a − e + e b, a e − e + e , . . . ,a e p − − e p − + e p − , a − e p − ba e p − − e p − ) . If e i − e i +1 + e i +2 (cid:54) = 0 for some i ∈ { , . . . , p − } , we have a non-trivial powerof a in one of the components of ψ ([ b, a, a ]) and we can argue as above to provethat ψ v ( γ ( G )) = G for a vertex v in the first level. On the other hand, if e i − e i +1 + e i +2 = 0 for all i = 1 , . . . , p −
3, then e = 2 e − e ,e = 2 e − e = 3 e − e , ... e p − = 2 e p − − e p − = ( p − e − ( p − e . Since e p − = e , the last equation implies that e = e , and then using all otherequations, we get that all components e i are equal to e . Thus the vector e isconstant, which is a contradiction. (cid:3) Acknowledgements.
The authors want to thank A. Tortora and G. Trausta-son for helpful discussions.
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E-mail address : [email protected] Dipartimento di Matematica, Universit`a di Salerno, Italy; Department of Math-ematics, University of the Basque Country UPV/EHU, Bilbao, Spain
E-mail address : [email protected] Department of Mathematical Sciences, University of Bath, Bath BA2 7AL,United Kingdom
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