aa r X i v : . [ m a t h . G R ] M a r Uniform Diameter Bounds in Branch Groups
Henry BradfordOctober 16, 2018
Abstract
Let G be either the Grigorchuk 2-group or one of the Gupta-Sidki p -groups. We give new upper bounds for the diameters of the quotients of G by its level stabilisers, as well as other natural sequences of finite-indexnormal subgroups. Our bounds are independent of the generating set,and are polylogarithmic functions of the group order, with explicit degree.Our proofs utilize a version of the profinite Solovay-Kitaev procedure, thebranch structure of G , and in certain cases, existing computations of thelower central series of G . Let G be a finite group, and S ⊆ G be a generating set. The diameter of G with respect to S is defined to be:diam( G, S ) = min { n ∈ N : B S ( n ) = G } ,where B S ( n ) is the (closed) ball of radius n about the identity in the wordmetric defined by S on G . The diameter of G , denoted diam( G ), is then themaximal value of diam( G, S ) as S ranges over all generating subsets of G . Inthis paper we give upper bounds for the diameters of natural families of finitequotients of certain branch groups . Theorem 1.1.
Let G be the Grigorchuk -group. Then: diam( G / Stab G ( n )) = O (cid:0) exp(log(35) n ) (cid:1) = O (cid:0) log | G : Stab G ( n ) | log(35) / log(2) (cid:1) . We shall define the sequence of level stabilisers
Stab G ( n ) for a group G acting on a rooted tree in subsection 2.2. Our proof makes extensive use of thedescription of the lower central series ( γ n ( G )) n of G given in [5], building on[26] (see also [3]). The results of these papers facilitate an explicit descriptionof the restriction to γ n ( G ) of the action of G on the binary rooted tree. Indeed,Theorem 1.1 is proved as a consequence of the following.1 heorem 1.2. diam( G /γ n ( G )) = O (cid:0) n log(35) / log(2) (cid:1) = O (cid:0) log | G : γ n ( G ) | log(35) / log(2) (cid:1) . Recall that log(35) / log(2) ≈ . p -groups. Theorem 1.3.
Let p be an odd prime. Let Γ ( p ) be the Gupta-Sidki p -group.Then: diam(Γ ( p ) / Stab Γ ( p ) ( n )) = O p (cid:0) exp(log( C p ) n ) (cid:1) = O p (cid:0) log | Γ ( p ) : Stab Γ ( p ) ( n ) | log( C p ) / log( p ) (cid:1) where C p = 3 · p − p ( p + 8) + 7 . Theorem 1.3 is a consequence of our next result. Let K be the derivedsubgroup of Γ ( p ) . Theorem 1.4.
Let C p is as in Theorem 1.3. diam(Γ ( p ) /K ( × p n ) ) = O p (cid:0) exp(log( C p ) n ) (cid:1) = O p (cid:0) log | Γ ( p ) : K ( × p n ) | log( C p ) / log( p ) (cid:1) . Here K ( × p n ) denotes the Cartesian product of p n copies of K . We definethe natural embeddings of the K ( × p n ) as finite-index normal subgroups of Γ ( p ) in subsection 2.2. For now suffice to say that there are inclusions K ( × p n ) ≤ Stab Γ ( p ) ( n + 1) and that given certain well-known bounds on the orders of therelevant groups, Theorem 1.3 quickly follows from these inclusions and Theorem1.4. In the case p = 3, we can exploit the description of the lower central seriesof Γ (3) given by Bartholdi [3] to also deduce the following. Theorem 1.5. diam(Γ (3) /γ n (Γ (3) )) = O (cid:0) n log(111) / log(1+ √ (cid:1) = O (cid:0) log | Γ (3) : γ n (Γ (3) ) | log(111) / log(3) (cid:1) . For the Gupta-Sidki 3-group therefore, our bounds for the diameter are poly-logarithmic in the order of the group, with degree log( C ) / log(3)= log(111) / log(3) ≈ . p grows, the degree of the polylogarithm grows,proportional to p/ log( p ). The implied constants in Theorems 1.1-1.5 may allbe explicitly computed from our proofs. G / Stab G ( n ) and Γ ( p ) / Stab Γ ( p ) ( n ) are transitive imprimitive permutationgroups on, respectively, 2 n and p n points. As such, Theorems 1.1 and 1.3 providenew examples of transitive subgroups of Sym( N ) ( N a power of a fixed prime)whose diameters are polynomially bounded in N . In the next subsection wewill further contextualise Theorems 1.1 and 1.3 within the existing literature ondiameters of permutation groups. G and Γ ( p ) are particularly famous membersof the class of branch groups, and have been extensively studied since theirintroduction, respectively in [16] and [19]. They will be defined precisely insubsection 2.2. .2 Background and Structure of the Paper G is one of the most exotic objects in geometric group theory: it is a finitelygenerated infinite 2-group, so provides a counterexample to the General Burn-side Problem; it is a group of intermediate growth , and indeed was the firstexample of such a group to be constructed; it is amenable but not elementaryamenable; it is a residually finite just-infinite group; it admits no faithful repre-sentation over any field, and every finite 2-group embeds into it as a subgroup[10]. Similarly, Γ ( p ) is a finitely generated infinite p -group; contains a copy ofevery finite p -group, and shares many of the other aforementioned propertieswith G , though its growth is the subject of ongoing discussion.An understanding of the broader class of branch groups , to which G and Γ ( p ) belong, has now become a crucial part of the toolkit of the modern geometric orprofinite group theorist. In one sense this is no surprise, bearing in mind Wil-son’s classification of just-infinite groups [29], in which branch groups comprisean important case (although historically, this may only be noted in hindsight,since the class of branch groups was not formally defined until some time afterWilson’s theorem [18]). What was perhaps less expected was the extraordinarilyrelevance branch groups would prove to have, to subjects as diverse as (but byno means limited to) decision problems, finite automata, spectral graph theory,fractal spaces, and exotic phenomena in the domains of word growth, subgroupgrowth and other asymptotic invariants of infinite groups [6].Seperate from, but roughly concurrent with these developments, interestwas growing in generation problems in various families of groups, including thesearch for good diameter bounds. From the start, particular attention was paidto upper bounds for permutation groups, owing to connections with problems intheoretical computer science. These included membership testing protocols incomputational group theory and complexity analysis of deciding solvability ofvarious combinatorial puzzles [13, 23], the most famous of which is of course theRubick’s cube. More recently intense interest in diameters of finite groups hasbeen renewed, motivated by connections with such diverse topics as expandergraphs, approximate groups, the Banach-Ruziewicz problem, Apollonian circlepackings, sum-product phenomena in fields and affine sieves. The modern studyof diameters of groups is therefore an extremely rich and diverse subject, and onewhich we cannot hope to fully capture here; we instead refer the interested readerto [20, 28] and references therein for an overview of the recent developments.Among the key tools to have featured in proofs of good upper bounds forthe diameters of finite groups is the Solovay-Kitaev procedure . This is a methodwhich was originally used in the context of compact complex Lie groups (whereit had applications to problems in quantum computer science) but which trans-lated readily to the setting of abstract or profinite groups. Given a group Γ and adescending sequence (Γ i ) i of finite-index normal subgroups, the Solovay-Kitaevprocedure proves, under suitable additional assumptions, an upper bound onthe diameters of the finite quotient groups Γ / Γ i by induction on the sequence(Γ i ) i . A crucial ingredient facilitating the induction step is that any elementof a later term in the sequence should be expressible as (or at least sufficientlylosely approximable by) the product of a small number of commutators of el-ements lying in earlier terms. Exactly what this means in practice will dependon the specific group Γ and sequence of subgroups with which we are working.In the above setting the Solovay-Kitaev procedure was first used by Gamburdand Shahshahani [14] to give a polylogarithmic upper bound on the diameterof SL ( Z /p n Z ). Their work was subsequently extended, first by Dinai [12] togroups of ( Z /p n Z )-points of other Chevalley groups, then by the author [8] tocongruence quotients of other p -adic analytic groups, F q [[ t ]]-analytic groups andthe Nottingham groups of finite fields.Producing the commutator expressions required to facilitate the inductionstep in the Solovay-Kitaev procedure requires fairly explicit computations ofcommutators in the subgroups Γ i . As such, it is very useful in practice forthe sequence (Γ i ) i to be highly “recurrent” in some sense, so that calculationscarried out at one level may be translated to others. In many of the examplesconsidered in [8], for instance, the group Γ has a naturally associated Lie algebraover a non-archimedean field, such that the Γ i may be identified with a descend-ing sequence of balls in the Lie algebra. Under this identification, computationsof commutators at different levels are simply “rescalings” of each other.The regular branch structure of G and Γ ( p ) provides a notion of “recurrence”of a different sort. These groups each have a finite-index normal subgroup K which naturally contains a direct product K ( × p ) of copies of itself, again as afinite-index normal subgroup (for G we take p = 2). We may therefore considerthe descending sequence Γ i = K ( × p i ) , and note that a commutator in Γ i +1 is a p -tuple of commutators in Γ i , thereby translating commutator calculations atthe top few levels down to all other levels.The best previous diameter bounds for quotients of branch groups make nouse of their branch structure, or indeed anything other than the fact that theyare transitive permutation groups. Babai and Seress obtained the following verygeneral result. Theorem 1.6 ([1] Theorem 1.4) . Let G be a transitive permutation group ofdegree N . Then: diam( G ) ≤ exp( C (log( N )) ) diam (cid:0) Alt( m ( G )) (cid:1) where C is an absolute constant, and m ( G ) is the maximal degree of an alter-nating composition factor of G . If Γ is a regular branch group acting on the k -ary rooted tree, then G = Γ / Stab Γ ( n ) has a natural transitive action on the n th level of the tree, soTheorem 1.6 is applicable, with N = k n and m ( G ) ≤ k . For Γ = G or Γ ( p ) ,Γ / Stab Γ ( n ) is a p -group of order Ω( p Ω( p n ) ) (see Lemma 2.9 and Corollary 4.1below), Theorem 1.6 yields:diam(Γ / Stab Γ ( n )) = O (cid:16) exp (cid:0) O ( n log( p ) ) (cid:1)(cid:17) = O (cid:16) exp (cid:0) O p (log log | Γ : Stab Γ ( n ) | ) (cid:1)(cid:17) here we take p = 2 for Γ = G ). Theorem 1.6 makes use of some deep machinery,including the Classification of Finite Simple Groups. Therefore our Theorems1.1-1.5 improve upon prior results in at least three ways:(i) They improve the diameter bound qualitatively, from a quasipolynomialfunction of log | G | to a polynomial one.(ii) They give explicit (and small) estimates for implied constants.(iii) They have self-contained, elementary and constructive proofs, which couldin principle be implemented in reasonable time on a computer.It is also noteworthy that Theorem 1.6 remains a key tool in the study ofother transitive permutation groups, which are consequently not known to havediameter less than exp( C (log( N )) ). The best known bound in this direction isthe following result of Helfgott and Seress. Theorem 1.7 ([21]) . Let G be as in Theorem 1.6. Then: diam( G ) ≤ exp( C (log( N )) log log( N )) . By Theorem 1.6, Theorem 1.7 can be immediately reduced to the case G =Alt( N ). The proof in this special case also uses Theorem 1.6 to facilitate animportant induction. It is a longstanding conjecture that Sym( N ) and Alt( N )in reality have diameter polynomial in N (that is, polylogarithmic in their order,like the groups studied in the present paper).A permutation group which provides a fascinating example intermediate be-tween Sym( N ) and the groups considered in Theorems 1.1 and 1.3, is the Sylow p -subgroup W n = Syl p (Sym( p n )) of Sym( p n ). W n was studied by Kaloujnine[22], who showed that it is isomorphic to the n -fold iterated regular wreathproduct C p ≀ C p ≀ · · · ≀ C p , which acts naturally on the (first n levels of the) p -aryrooted tree. Although the inverse limit Γ = lim ←− n W n of the W n is a regularbranch pro- p group, it appears to be resistant to the Solovay-Kitaev procedure(for instance, Γ is not finitely generated as a topological group; by contrast,every version of the profinite Solovay-Kitaev procedure known to the authorproves finite generation as a byproduct). It is however likely that the methodsof this paper are applicable to other residually nilpotent branch groups. Thisshould be a topic of further study.The paper is structured as follows. Section 2 is devoted to preliminarymaterial: in subsection 2.1 we lay out the consequences of the standard com-mutator identities upon which the profinite Solovay-Kitaev procedure is based,and illustrate, by means of an example, their relevance to diameter bounds. Insubsection 2.2 we recall some basic material on group actions on regular rootedtrees and the class of (regular) branch groups, define the Grigorchuk group G and the Gupta-Sidki p -groups Γ ( p ) and give some basic properties. In Section 3we prove Theorem 1.2, and deduce Theorem 1.1. These proofs will be based inpart upon prior results on the structure of the lower central series of G (takenfrom [3, 5]), which we state there. In Section 4 we prove Theorem 1.4 and de-duce Theorem 1.3. In Section 5 we recall some results from [3] on the structuref the lower central series of Γ (3) and using them, deduce Theorem 1.5 fromTheorem 1.4. In Section 6 we comment on implications of our diameter boundsfor spectral gap and mixing times of random walks on Cayley graphs. Finallyin Section 7 we make some remarks on the relationship between the growth ofan infinite group and the diameters of its finite quotients. The proofs of Theorems 1.2 and 1.4 (from which our other results are deduced)will be via a variant of the profinite Solovay-Kitaev Procedure developed in [8],to which we refer the reader for further background. Given a group G and adescending sequence of finite-index normal subgroups, the Procedure providesan approach to proving upper bounds on the diameters of the correspondingfinite quotient groups, and relies on two key ingredients, both concerning thebehaviour of commutator words within the group. The first ingredient encap-sulates the intuitive notion that, given two specified elements g, h ∈ G and two“good approximations” ˜ g, ˜ h ∈ G , the commutator [˜ g, ˜ h ] of the approximations isa good approximation to the commutator [ g, h ] of the original elements. Whatis perhaps not so intuitive is that in many situations, [˜ g, ˜ h ] approximates [ g, h ]much more closely than ˜ g and ˜ h did g and h . This idea is made precise in thefollowing Lemma, which is an immediate consequence of standard commutatoridentities. Lemma 2.1.
Let G , G , H , H ⊳ G , with G ≥ G , H ≥ H . For all g i ∈ G i , h i ∈ H i ( i = 1 , ), [ g g , h h ] ≡ [ g , h ] mod [ G , H ][ G , H ] .Proof. We compute directly:[ g g , h h ] = [ g , h ][ g , h ][[ g , h ] , h ][[ g , h h ] , g ][ g , h h ]and all terms other than [ g , h ] lie in [ G , H ][ G , H ].A particularly useful special case of this lemma is the following. Corollary 2.2.
Let ( K i ) ∞ n =1 be a descending sequence of normal subgroups of G . Suppose that, for all m, n ∈ N , [ K m , K n ] ⊆ K m + n . Let m , m , n , n ∈ N ,with m ≤ m , n ≤ n , and let g i ∈ K m i , h i ∈ K n i ( i = 1 , ). Then: [ g g , h h ] ≡ [ g , h ] mod K min( m + n ,m + n ) .Proof. Set G i = K m i and H i = K n i in Lemma 2.1, for i = 1 , Lemma 2.3.
Let G be any group. Then for all m, n ∈ N , γ n ( G ) , γ m ( G )] ⊆ γ m + n ( G ) . Lemma 2.3 will be used extensively and without further comment in thesequel, and in particular throughout Section 3.What is the relevance of the preceding discussion to diameters? As weintimated in the introduction, given a group Γ and a descending sequence (Γ i ) i ,the Solovay-Kitaev procedure requires the existence of an approximation toelements of a deeper term Γ j in the sequence, by commutators [ g, h ] of elements g, h lying in a higher term Γ i . We may assume by induction that we haveapproximations ˜ g, ˜ h to g, h up to an error lying in the intermediate term Γ k ,where ˜ g and ˜ h are short words in a generating set. Substituting ˜ g and ˜ h intoour commutator expression, we obtain approximations [˜ g, ˜ h ] to elements of Γ j by short words. Lemma 2.1 gives us some control over the fidelity of theseapproximations. Let us illustrate this with an example. Example 2.4.
Let Γ be a group, let Γ ≥ Γ ≥ Γ ≥ Γ be finite-index normalsubgroups of Γ, and suppose: [Γ , Γ ] ≤ Γ . (1)Second, suppose that for any k ∈ Γ , there exist g, h ∈ Γ such that:[ g, h ] k − ∈ Γ . (2)Now let S ⊆ Γ, and suppose the image of S in Γ / Γ is a generating set. Let d = diam(Γ / Γ , S ), so that: Γ = Γ B S ( d ) (3)(here we abuse notation slightly, denoting also by “ S ” the image of S moduloany of the Γ i ).We wish to bound diam(Γ / Γ , S ). That is, given k ∈ Γ we seek a short word w in S such that kw − ∈ Γ . First suppose that k ∈ Γ . Let g, h ∈ K beas in (2). By (3), there exist ˜ g, ˜ h ∈ B S ( d ) such that g − ˜ g, h − ˜ h ∈ Γ . Setting G = H = Γ , G = H = Γ , g = g , h = h , g = g − ˜ g , h = h − ˜ h inLemma 2.1, and applying (1),[˜ g, ˜ h ] ≡ [ g, h ] ≡ k mod Γ .Since [˜ g, ˜ h ] ∈ B S (4 d ), we have Γ ⊆ Γ B S (4 d ). Applying (3) again, Γ =Γ B S (5 d ), so diam(Γ / Γ , S ) ≤ d .Although the hypotheses (1) and (2) may seem somewhat artificial in thecontext of this abstract example, in practice many groups satisfy these con-ditions, or variants thereof. Modifications to the method of Example 2.4 arehowever sometimes necessary, or desirable. For instance the elements g and h in Example 2.4 were taken to lie at the same level of the subgroup chain,whereas for many groups, including the branch groups studied in this paper,good commutator expressions will involve elements lying at different levels. .2 Groups Acting on Regular Rooted Trees Let us set up some basic notation and definitions concerning the class of groupsto be studied in the sequel. All of the material here (and much, much morebesides) is covered in [6] and [10] Chapter VIII.Let A be a finite set, and let A ∗ be the set of formal positive words on thealphabet A . We may partially order A ∗ via the prefix relation ≤ , where for u, v ∈ A ∗ , u ≤ v iff there exists w ∈ A ∗ such that v = uw .Geometrically, we may regard A ∗ as the set of vertices of a regular rootedtree T A : the root vertex is identified with the empty word, and every vertex v is joined by an edge to its |A| children va , a ∈ A . Under this identification, theset A n of words of length n is precisely the sphere of radius n in T A about theroot vertex, known as the n th level set .The automorphism group Aut( T A ) is the set of permutations of A ∗ preservingthe prefix relation. Geometrically this just the group of graph automorphismsof the tree T A . For the valence of every vertex of T A is |A| + 1 except for theroot vertex (which has valence |A| ), so every automorphism of T A fixes the rootvertex, and hence preserves the level sets. The kernel of the action of Aut( T A ) onthe n th level set A n will be called the n th level stabiliser and denoted Stab( n );it is naturally isomorphic to Aut( T A ) ( ×|A| n ) . If Γ ≤ Aut( T A ) we write Stab Γ ( n )for Γ ∩ Stab( n ), though in general we cannot say more about the structure ofStab Γ ( n ) than that Stab Γ ( n ) is isomorphic to a subgroup of Aut( T A ) ( ×|A| n ) .For any φ ∈ Aut( T A ), there exists a unique σ φ ∈ Sym( A ) such that for any x ∈ A , there exists a unique φ x ∈ Aut( T A ) such that: φ ( xw ) = σ φ ( x ) φ x ( w ), for all w ∈ A ∗ .The induced map ψ : φ ( φ x ) x ∈A · σ φ gives an isomorphismAut( T A ) → Aut( T A ) ≀ Sym( A ). Note that the level stabilisers may be describedrecursively by Stab(0) = Aut( T A ) and Stab( n + 1) = ψ − (Stab( n ) ( ×|A| ) ) for n ∈ N .Of particular interest among the subgroups of Aut( T A ) are those whoseaction on A ∗ is branch . Our characterization of such groups is based on thatappearing in [3]. Definition 2.5.
Let Γ ≤ Aut( T A ) . Γ is (regular) branch if:(i) The action of Γ on A is transitive;(ii) ψ (Stab Γ (1)) ≤ Γ ( ×|A| ) ;(iii) Γ has a finite-index subgroup K such that K ( ×|A| ) ≤ ψ ( K ) .We will simply say that a group Γ branches over K when the alphabet A andthe action of Γ on A ∗ is clear. For the sake of uncluttered notation we will allow ourselves to supress themap ψ from expressions and identify subgroups of Γ with their image under ψ ,so for instance we may (abuse notation somewhat and) speak of K ( ×|A| ) as asubgroup of K ; Stab Γ ( n ) as a subgroup of Γ ( ×|A| n ) and so on.n truth, branch groups form a much broader class than those groups coveredby Definition 2.5 (see for instance [6]), but this more restricted setting will bemost convenient for our purposes.We now define the specific branch groups which are the subject of Theorems1.1-1.5. -Group Let A = { , } and write T A = T . The Grigorchuk -group (sometimes knownas the first Grigorchuk group ) is the subgroup G of Aut( T ) generated by thefour automorphisms a, b, c, d , defined by: a (0 w ) = 1 w ; a (1 w ) = 0 w ; b (0 w ) = 0 a ( w ); b (1 w ) = 1 c ( w ); c (0 w ) = 0 a ( w ); c (1 w ) = 1 d ( w ); d (0 w ) = 0 w ; d (1 w ) = 1 b ( w ).In other words, a swaps the subtrees rooted at 0 and 1, while b, c, d ∈ Stab G (1)are defined recursively via: b = ( a, c ) , c = ( a, d ) , d = (1 , b ).An easy induction on the levels shows that: a = b = c = d = 1 , bc = cb = d, cd = dc = b, bd = db = c . (4)Let x = abab ∈ G and let K = h x i G ⊳ G . We will require the following basicfacts in the sequel. Proposition 2.6 ([10] Chapter VIII) . Let G , K , x be as above.(i) G branches over K ;(ii) G /K ∼ = D × C ;(iii) K/K ( × ∼ = C , generated by x . Lemma 2.7. K ( × m ) ⊳ G for all m ≥ .Proof. We proceed by induction on m , the base case m = 0 holding by definition.For m ≥
1, we verify that K ( × m ) is preserved under conjugation by thegenerators a, b, c, d . This is the case for a , which simply permutes the factors inthe direct product. b, c and d preserve the decomposition K ( × m ) = ( K ( × m − ) ) ( × , and act oneach K ( × m − ) -factor as a, b, c or d . The result follows by induction. Lemma 2.8.
For all m ≥ , K ( × m ) ≤ Stab G ( m + 1) .Proof. Note that x = abab ∈ Stab G (1). The result is now immediate fromProposition 2.6 (iii). emma 2.9. For all n ≥ , | G : Stab G ( n ) | ≥ n − +1 .Proof. It suffices to prove that | Stab G ( m ) : Stab G ( m + 1) | ≥ m − for all m ∈ N . x = abab ≡ ( a, a ) mod Stab Aut( T ) (2), so as ǫ ranges over { , } m , theelements ( x ǫ i ) m i =1 are all distinct modulo Stab G ( m + 1), and lie in Stab G ( m ) byLemma 2.8. p -Groups Fix an odd prime p , let A = { , , . . . , p − } and write T A = T p . The Gupta-Sidki p -group is the subgroup Γ ( p ) of Aut( T p ) generated by the two automorphisms a, b , where a is defined by: a ( iw ) = ( i + 1) w for 0 ≤ i ≤ p − a (( p − w ) = 0 w (so that a cyclically permutes the level-1 subtrees) and b ∈ Stab
Aut( T p ) (1) isdefined recursively via: b = ( a, a − , , . . . , , b )so that both a and b have order p . Note that a reduced word in a and b corre-sponds to an element of Stab Γ ( p ) (1) iff the number of occurences of a (countedwith signs) is congruent to 0 modulo p . Thus Stab Γ ( p ) (1) = h b i Γ ( p ) .Let K = [Γ ( p ) , Γ ( p ) ] be the derived subgroup of Γ ( p ) . Note that K ≤ Stab Γ ( p ) (1). Let x = [ a, b ] ∈ K , and for 1 ≤ i ≤ p −
2, define x i +1 = [ a, x i ] ∈ K . The following computations were made in [15], general-ising results from [27] for Γ (3) . Proposition 2.10 ([15] Proposition 2.2) . Let B = Stab Γ ( p ) (1) . Then:(i) Γ ( p ) /K ∼ = C p × C p , with basis Ka, Kb ;(ii) B ′ = K ( × p ) ;(iii) Γ ( p ) /B ′ ∼ = C p ≀ C p . From this we have an explicit description of the branch structure of Γ ( p ) . Corollary 2.11. (i) Γ ( p ) branches over K ;(ii) K/K ( × p ) ∼ = C ( × ( p − p , with basis x , . . . , x p − .Proof. (i) From Proposition 2.10 (ii), K ( × p ) = B ′ ≤ Γ ′ ( p ) = K .(ii) We defer this to Section 4, where we introduce additional notation whichwill be convenient to use in the proof.he following observation will be key to the deduction of Theorem 1.3 fromTheorem 1.4. Lemma 2.12.
For all m ∈ N , K ( × p m ) ≤ Stab Γ ( p ) ( m + 1) .Proof. It clearly suffices to check that K ≤ Stab Γ ( p ) (1). This is so because K = [Γ ( p ) , Γ ( p ) ] and Γ ( p ) / Stab Γ ( p ) (1) is abelian. In this Section we prove Theorem 1.2 and deduce Theorem 1.1. Before embark-ing on the proof of our diameter bounds, we marshal some facts about the lowercentral series of G .Recall that for any group G , the degree deg( g ) of g ∈ G is given by g ∈ γ deg( g ) ( G ) \ γ deg( g )+1 ( G ), with the convention that all g ∈ T ∞ n =1 γ n ( G ) havedegree ∞ (though in G , the latter situation never arises for g = 1: G is a residu-ally finite 2-group, so is residually 2-finite, and in particular, T ∞ n =1 γ n ( G ) = { } ).For any g ∈ K , write ( g ) = ( g, , ( g ) = ( g, g − ) ∈ K ( × . Theorem 3.1 ([3, 5, 26]) . Let X , . . . , X n ∈ { , } . Then: deg X · · · X n ( x ) = 1 + n X i =1 X i i − + 2 n ; deg X · · · X n ( x ) = 1 + n X i =1 X i i − + 2 n +1 . Theorem 3.2 ([3, 5, 26]) . For all m ≥ , | G : γ m +1 ( G ) | = 2 m − . To avoid cluttered notation, in this section we may write γ n = γ n ( G ). Thespecific consequences of Theorem 3.1 to be used in the proof of Theorem 1.2 areas follows. Corollary 3.3. γ m +2 m − +1 ≤ K ( × m ) ≤ γ m +1 . To be more precise, we have the following estimates.
Corollary 3.4.
For all δ ∈ (cid:0) { , } n − (cid:1) \ { } ,(i) n + 1 ≤ deg (cid:0)(cid:0) ( x, , , δ i (cid:1) n − i =1 (cid:1) ≤ n + 2 n − ;(ii) n + 2 n − + 1 ≤ deg (cid:0)(cid:0) ( x, , x, δ i (cid:1) n − i =1 (cid:1) ≤ n + 2 n − ;(iii) n + 2 n − + 1 ≤ deg (cid:0)(cid:0) ( x, x, , δ i (cid:1) n − i =1 (cid:1) ≤ n + 3 · n − ;(iv) n + 3 · n − + 1 ≤ deg (cid:0)(cid:0) ( x, x, x, x ) δ i (cid:1) n − i =1 (cid:1) ≤ n +1 ;v) Specifically, deg (cid:0)(cid:0) ( x, x, x, x ) (cid:1) n − i =1 (cid:1) = 2 n +1 ;(vi) n + 1 ≤ deg (cid:0)(cid:0) ( x , δ i (cid:1) n − i =1 (cid:1) ≤ n + 2 n − ;(vii) n + 2 n − + 1 ≤ deg (cid:0)(cid:0) ( x , x ) δ i (cid:1) n − i =1 (cid:1) ≤ n + 2 n − ;(viii) Specifically, deg (cid:0)(cid:0) ( x , x ) (cid:1) n − i =1 (cid:1) = 2 n + 2 n − . We now produce our commutator approximations to elements lying suffi-ciently deep in ( γ n ) n . The following identities are verified by direct computa-tion. Lemma 3.5. (i) (cid:2) x, ( x, (cid:3) = (cid:0) x − , , , (cid:1) .(ii) (cid:2) x, ( x, x ) (cid:3) = (cid:0) x − , , , (1 , x − ) x (cid:1) .(iii) (cid:2) x , ( x, (cid:3) = (cid:0) x − , x, , (cid:1) .(iv) (cid:2) x , ( x, x − ) (cid:3) = (cid:0) x − , x, ( x − , x − , (1 , x − ) x (cid:1) .Proof. We prove (i) and leave the verifications of the other identities, which aresimilar, as an exercise. First note that x = abab = ( ca, ac ). Thus: (cid:2) x, ( x, (cid:3) = (cid:0) [ ca, x ] , (cid:1) Now, using (4) we have: [ ca, x ] = ac ( ac, ca ) ca ( ca, ac )= a ( ca, bad ) a ( ca, ac )= ( bad, ca )( ca, ac )= ( baba, x − , K ( × , modulo K ( × ,by commutators of elements of K and K ( × . For deeper subgroups K ( × n ) , weexpress elements as vectors of elements in K ( × , and produce a commutatorapproximation by applying Lemma 3.5 to each term of the vector. The expres-sions we obtain can be related to the lower central series by using Theorem 3.1and Corollaries 3.3 and 3.4 to estimate the degrees of the elements occuring. Insummary we have the following Proposition. Proposition 3.6.
Let c : G × G → G be given by c ( g, h ) = [ g, h ] . Let m ≥ .Then:(i) The restriction of c to γ m − × γ m − descends to a well-defined map: ¯ c m : ( γ m − /γ m +1 ) ( × → G /γ m +2 m − +1 hose image contains K ( × m ) /γ m +2 m − +1 .(ii) The restriction of c to γ m − +2 m − × γ m − +2 m − descends to a well-definedmap: ¯¯ c m : ( γ m − +2 m − /γ m +2 m − +1 ) ( × → G /γ m +1 +1 whose image contains γ m +2 m − +1 /γ m +1 +1 .Proof. The well-definedness of ¯ c m and ¯¯ c m is an immediate consequence of Corol-lary 2.2. It therefore suffices to check the images of the maps contain the spec-ified subgroups.For (i), note that by Corollary 3.4, every element of K ( × m ) /γ m +2 m − +1 isrepresented by (cid:0) ( x δ i + ǫ i , , x ǫ i , (cid:1) m − i =1 for some δ i , ǫ i ∈ { , } . By Lemma 3.5(i) and (ii), (cid:2) x, ( x, (cid:3) · (cid:0) x, , , (cid:1) − = (cid:0) x , , , (cid:1) − (cid:2) x, ( x, x ) (cid:3) · (cid:0) x, , x, (cid:1) − ≡ (cid:0) , , x, x (cid:1)(cid:0) x , , x , (cid:1) − mod K ( × ) .By Corollary 3.4 (iii) and (iv), for any ( β i ) m − i =1 ∈ { , } m − , (cid:0) (1 , , x, x ) β i (cid:1) m − i =1 ∈ γ m +2 m − +1 ,by Corollary 3.4 (vi), for any ( β i ) m − i =1 ∈ { , } m − , (cid:0) ( x , , , β i (cid:1) m − i =1 , (cid:0) ( x , , x , β i (cid:1) m − i =1 ∈ γ m +2 m − +1 ,and from Corollary 3.3, K ( × m +1 ) ⊆ γ m +1 +1 . Hence, for any ( δ i ) m − i =1 ,( ǫ i ) m − i =1 ∈ { , } m − , (cid:0) ( x δ i + ǫ i , , x ǫ i , (cid:1) m − i =1 ≡ (cid:2) ( x ) m − i =1 , (( x, δ i ) m − i =1 (cid:3) · (cid:2) ( x ) m − i =1 , (( x, x ) ǫ i ) m − i =1 (cid:3) mod γ m +2 m − +1 .Using Corollary 3.4 to estimate the degrees of ( x ) m − i =1 , (( x, δ i ) m − i =1 and (( x, x ) ǫ i ) m − i =1 ,and by the standard identity [ a, bc ] = [ a, c ][ a, b ][[ a, b ] , c ], we deduce: (cid:0) ( x δ i + ǫ i , , x ǫ i , (cid:1) m − i =1 ≡ (cid:2) ( x ) m − i =1 , (( x δ i + ǫ i , x ǫ i )) m − i =1 (cid:3) mod γ m +2 m − +1 .(5)For (ii), we see similarly by Corollary 3.4 that every element of γ m +2 m − +1 /γ m +1 +1 is represented by (cid:0) ( x δ i + ǫ i , x δ i + ǫ i , x ǫ i , x ǫ i ) (cid:1) m − i =1 for some δ i , ǫ i ∈ { , } . ByLemma 3.5 (iii) and (iv), (cid:2) x , ( x, (cid:3) · (cid:0) x, x, , (cid:1) − = (cid:0) x , , , (cid:1) − (cid:2) x , ( x, x − ) (cid:3) · (cid:0) x, x, x, x (cid:1) − ≡ (cid:0) x , , x , (cid:1) − mod K ( × ) .sing Corollary 3.3 and Corollary 3.4 (vi) once again as in (i), we have that forany ( δ i ) m − i =1 , ( ǫ i ) m − i =1 ∈ { , } m − , (cid:0) ( x δ i + ǫ i , x δ i + ǫ i , x ǫ i , x ǫ i ) (cid:1) m − i =1 ≡ (cid:2) ( x ) m − i =1 , (( x, δ i ) m − i =1 (cid:3) · (cid:2) ( x ) m − i =1 , (( x, x − ) ǫ i ) m − i =1 (cid:3) mod γ m +2 m − +1 .As before, we apply the commutator identity for products, using the estimate ofthe degrees of ( x ) m − i =1 , (( x, δ i ) m − i =1 and (( x, x − ) ǫ i ) m − i =1 from Corollary 3.4,and deduce: (cid:0) ( x δ i + ǫ i , x δ i + ǫ i , x ǫ i , x ǫ i ) (cid:1) m − i =1 ≡ (cid:2) ( x ) m − i =1 , (( x δ i + ǫ i , x − ǫ i )) m − i =1 (cid:3) mod γ m +1 +1 .(6)Using Proposition 3.6 we may approximate any tuple consisting of x s and1s by a commutator. To express arbitrary tuples in K ( × m − ) /K ( × m ) we mustalso find approximations for tuples consisting of x s and 1s. Here we will divergeslightly from our overall strategy of approximating elements by commutators,since it appears far more natural to express such tuples as squares. Proposition 3.7.
The squaring map s : g g on G induces a surjection: s m : K ( × m − ) /γ m +1 → γ m +1 /K ( × m ) for all m ≥ .Proof. It is immediate from Lemma 2.6 and Corollaries 3.3 and 3.4 that K ( × m − ) /γ m +1 and γ m +1 /K ( × m ) are elementary abelian 2-groups, each ele-ment of the latter being represented by a vector:( x δ i ) m − i =1 = s (( x δ i ) m − i =1 ),as δ ranges over { , } m − . Since ( x δ i ) m − i =1 ∈ K ( × m − ) , s induces a surjection K ( × m − ) → γ m +1 /K ( × m ) .Now let a ∈ K ( × m − ) , b ∈ γ m +1 . We have: s ( ab ) = s ( a )[ a, b − ] s ( b ).But s ( γ m +1 ) ⊆ K ( × m ) (as noted above) and:[ K ( × m − ) , γ m +1 ] ⊆ K ( × m ) by Corollary 3.3 (applied to both K ( × m − ) and K ( × m ) ). Thus s m is indeedwell-defined.We now come to the heart of the proof of our diameter bound: using Propo-sitions 3.6 and 3.7, we show that if a symmetric subset X ⊆ G contains anapproximation to every element of G up to an error lying in γ m +1 , then everyelement of G is approximated, up to an error in the (much smaller) subgroup γ m +1 +1 , by a short word in X . roposition 3.8. Let m ≥ and let X ⊆ Γ be a symmetric subset such that: Xγ m +1 = G . (7) Then: X γ m +1 +1 = G . (8) Proof.
First, note that Proposition 3.7 immediately implies: X K ( × m ) ⊇ γ m +1 . (9)Second, we combine (7) and Proposition 3.6 (ii) with Corollary 2.2 to con-clude: K ( × m ) ⊆ X γ m +2 m − +1 . (10)Taking stock of what we have thus far, (7), (9) and (10) combine to give: X γ m +2 m − +1 = G . (11)Finally, we combine (11) and (6) with Corollary 2.2 and conclude: γ m +2 m − +1 ⊆ X γ m +1 +1 . (12)The required conclusion (8) is now immediate from (11) and (12). Proof of Theorem 1.2.
Let S ⊆ G /γ n ( G ) be a generating set. If n ≤ G /γ n ( G ) , S ) ≤ | G : γ n ( G ) | is bounded by an absolute constant ˜ C . Otherwise let ˜ S ⊆ G be any subset whoseimage in G /γ n ( G ) is S , and let m ∈ N be such that 2 m − + 1 < n ≤ m + 1.Then B ˜ S ( ˜ C ) γ ( G ) = G , and by repeated application of Proposition 3.8, B ˜ S (35 m − ˜ C ) γ m +1 ( G ) = G so that diam( G /γ n ( G ) , S ) ≤ m − ˜ C ≪ n log(35) / log(2) .The result now follows from Theorem 3.2. Remark 3.9.
Note that the above proof facilitates straightforward computationof the implied constant from the statement of Theorem 1.2.Proof of Theorem 1.1.
By Lemma 2.8 and Corollary 3.3, γ n +1 +1 ( G ) ≤ Stab G ( n + 1)so: diam( G / Stab G ( n + 1)) ≤ diam( G /γ n +1 +1 ( G )).The result is now immediate from Theorem 1.2, Lemma 2.9 and Theorem 3.2. emark 3.10. It is very likely that detailed knowledge of the lower centralseries of G is not required to prove Theorem 1.1. Rather, one could give adirect proof of a diameter bound for G /K ( × n ) and deduce Theorem 1.1 fromthis, much as we do for the Gupta-Sidki groups in the following Section. Thereason for organizing the proof of Theorem 1.1 as it appears here is historical.Theorem 1.2 was the first of our results to be proved, followed be a directproof of Theorem 1.5, using the results of [3] on the lower central series of Γ (3) .The case p = 3 of Theorem 1.3 was then deduced from Theorem 1.5 (much asTheorem 1.1 follows from Theorem 1.2). The (arguably more natural) proofof Theorem 1.3 from Theorem 1.4 was a later response to the need to avoidassuming knowledge of the lower central series of Γ ( p ) in proving Theorem 1.3for higher p (to the author’s knowledge, the lower central series of Γ ( p ) has notbeen computed for p ≥ Remark 3.11.
Proposition 3.7 hints at the possibility of an alternative ap-proach to proving diameter bounds for sequences of groups, following the samebroad lines as the Solovay-Kitaev procedure employed here and in [8], but us-ing power-words instead of commutator words. This will be explored furtherelsewhere [9].
In this section we prove Theorem 1.4 and, from it, Theorem 1.3. For the re-mainder of the section we shall write Γ for Γ ( p ) . Any assumptions on the prime p in what follows will be made explicit in the appropriate place.The following notation will be useful in the sequel: for any g ∈ Aut( T A ),let ( g ) = ( g, , . . . , ∈ Aut( T A ) ( × p ) , and for 0 ≤ j ≤ p −
1, let( j + )( g ) = [ a, j ( g )]. Hence for 0 ≤ j ≤ p − j ( g ) i = (cid:8) g α j,i ≤ i ≤ j + 11 otherwise , where α j,i = ( − i +1 (cid:18) ji − (cid:19) .Note that α p − ,i ≡ p for all 1 ≤ i ≤ p . It follows that:( p − )( a ) = ( a, . . . , a ), ( p − )( b ) = ( b, . . . , b )(since a, b have order p ) and:( p − )( g ) ≡ ( g, . . . , g ) mod K ( × p ) for any g ∈ K (since K ≤ Stab Γ (1) and Γ /K has exponent p , so too does K/K ( × p ) ).Note also that in this notation, b = ( a ) ( b ) a − = ( b ) a − ( a ).It follows from the definition of the elements x i that for 1 ≤ i ≤ p − x i (cid:0) ( i + )( a ) i ( b ) a − (cid:1) − ∈ K ( × p ) (13)nd in particular: x p − ( a, . . . , a ) − ∈ (Stab Γ (1)) ( × p ) (14) x p − ( b, . . . , b ) − ∈ K ( × p ) (15)These observations facilitate the completion of: Proof of Corollary 2.11 (ii).
Since K ≤ Stab Γ (1), K/K ( × p ) is an elementaryabelian p -group. Moreover, from Proposition 2.10 (i) and (iii) we have | K : K ( × p ) | = p p − , so it suffices to check that the images of x , . . . , x p − in K/K ( × p ) are linearly independent.Embedding Stab Γ (1) into Γ ( × p ) (via ψ ), we have an induced embedding of K/K ( × p ) into (Γ /K ) ( × p ) . By Proposition 2.10 (i),(Γ /K ) ( × p ) = h Ka, Kb i ( × p ) ∼ = C ( × p ) p .From (13), the image of x i in (Γ /K ) ( × p ) is v i = ( i + )( Ka ) i ( Kb ) a − . For all1 ≤ i ≤ p − v i has non-zero Ka -component in its ( i + 2)th entry, but zero Ka -component in its j th entry for all j ≥ i +3, so that v i is linearly independentof h v , . . . , v i − i .Similarly, for all 1 ≤ i ≤ p − v i has zero Kb -component in its ( p − v p − has non-zero Kb -component there, so v p − is independentof h v , . . . , v p − i . Corollary 4.1.
For all n ≥ , | Γ : Stab Γ ( n ) | ≥ p ( p − p n − − .Proof. It suffices to check | Stab Γ ( m ) : Stab Γ ( m + 1) | ≥ p ( p − p m − − for all m ≥ Γ ( m ) into Γ ( × p m ) (via repeated application of ψ ) we havean induced embedding:Stab Γ ( m ) / Stab Γ ( m + 1) ֒ → (Γ / Stab Γ (1)) ( × p m ) ∼ = C ( × p m ) p (with each (Γ / Stab Γ (1))-factor generated by Stab Γ (1) a ). From (13), the imageof x i ∈ Stab Γ (1) in (Γ / Stab Γ (1)) ( × p ) is ( i + )(Stab Γ (1) a ).Arguing as in the proof of Corollary 2.11 (ii), the elements: (cid:16) p − Y j =1 x λ i,j j (cid:17) p m i =1 are distinct modulo Stab Γ ( m + 1) as the p m − × ( p −
2) coefficients λ i,j varyover { , , . . . , p − } , and the required result follows.We also introduce some further normal subgroups which will be useful “place-holders” for our induction in the proof of Theorem 1.4. For 1 ≤ i ≤ p , let: L i = (cid:10) x i , . . . , x p − , K ( × p ) (cid:11) ≤ K with L p = K ( × p ) by convention). For 0 ≤ i ≤ p , let: K ( × p ) i = (cid:10) i ( x ) , . . . , ( p − )( x ) , L ( × p )2 (cid:11) ≤ K ( × p ) ; K ( × p n +1 ) i = ( K ( × p ) i ) ( × p n ) (with the convention K ( × p ) p = L ( × p )2 ). We thus have descending chains of sub-groups: L ( × p n )2 = K ( × p n ) p ≤ K ( × p n ) p − ≤ . . . ≤ K ( × p n ) ≤ K ( × p n ) = K ( × p n ) and: K ( × p n +1 ) = L ( × p n ) p ≤ L ( × p n ) p − ≤ . . . ≤ L ( × p n )2 . Lemma 4.2.
The following are normal in Γ ( p ) , for all n ∈ N :(i) K ( × p n ) ;(ii) L ( × p n ) i , for ≤ i ≤ p − ;(iii) K ( × p n +1 ) i , for ≤ i ≤ p − .Proof. First, by induction we reduce to the case n = 0. For let H = K ( × p n ) , K ( × p n +1 ) i or L ( × p n ) i . If H ⊳ Γ ( p ) , then H ( × p ) is normalised by a (which permutesthe factors) and by b (which acts on each factor as a ± or b ).Now certainly K ⊳ Γ ( p ) , so we have (i). In the other cases, we check thatconjugates of a generating set for the subgroup, by the generators of Γ ( p ) , lie inthe subgroup.For (ii), consider the conjugation action of Γ ( p ) on K/K ( × p ) . We have x ai = x i x − i +1 for 1 ≤ i ≤ p − x ap − ≡ x p − mod K ( × p ) and x bi ≡ x i mod K ( × p ) for 1 ≤ i ≤ p −
1. Thus L i ⊳ Γ ( p ) .For (iii), consider the conjugation action of Γ ( p ) on K ( × p ) /L ( × p )2 .We have i ( x ) a = i ( x )( i + )( x ) − for 1 ≤ i ≤ p − p − )( x ) a ≡ ( p − )( x ) mod L ( × p )2 and i ( x ) b ≡ i ( x ) for 1 ≤ i ≤ p − K ( × p ) i ⊳ Γ ( p ) .To apply Lemma 2.1 in our induction, we will need certain commutatorsof elements in Γ to lie in a sufficiently deep subgroup. As such, we note thefollowing. Lemma 4.3. (i) [Stab Γ (1) , Stab Γ (1)] ≤ K ( × p ) ;(ii) [Stab Γ (1) , K ( × p ) ] ≤ L ( × p )2 .Proof. For (i), note that any element of Stab Γ (1) is a p -tuple of elements of Γ,so any commutator of such is a p -tuple of elements of [Γ , Γ] = K .Likewise in (ii), every element of [Stab Γ (1) , K ( × p ) ] is a p -tuple of elementsof [Γ , K ]. K/L is generated by x , and Γ is generated by a and b , so, since L ⊳ K , it suffices to check that [ a, x ] , [ b, x ] ∈ L (the former by definition;the latter by (i)).e now describe our approximations to elements of Γ by commutators.These are split between the next four propositions, which are closely reminis-cent of Proposition 3.6 in our proof for Grigorchuk’s group. The first of theseallows us to approximate elements of K ( × p ) up to an error in K ( × p ) . We re-mark that this is the only point in the proof of Theorem 1.4 at which differentarguments are required according to the value of p . To be precise, we exhibitone construction of an approximation by commutators which is valid for p ≥ p = 5 and one for p = 3.Let c : Γ × Γ → Γ be given by c ( g, h ) = [ g, h ]. Proposition 4.4.
The restriction of c to L p − × K ( × p ) descends to a well-defined map: ¯ c : ( L p − /K ( × p ) ) × ( K ( × p ) /K ( × p ) ) → Γ /K ( × p ) whose image contains K ( × p ) /K ( × p ) .Proof. Observe that by Lemma 2.1, for g ∈ L p − , h ∈ K ( × p ) and g , h ∈ K ( × p ) we have:[ g g , h h ][ g , h ] − ∈ (cid:2) L p − , K ( × p ) (cid:3) .But L p − is generated by K ( × p ) and x p − , both of which lie in Stab Γ (1) ( × p ) (the latter since x p − is congruent modulo K ( × p ) to ( b, . . . , b )). Hence: (cid:2) L p − , K ( × p ) (cid:3) ⊆ (cid:2) Stab Γ (1) , K ( × p ) (cid:3) ( × p ) ⊆ L ( p )2 ⊆ K ( × p ) (by Lemma 4.3 (ii)).Thus ¯ c is indeed well-defined.We now establish that the image of ¯ c contains K ( × p ) /K ( × p ) . First suppose p ≥
7. We have: x a = ( b, b − a, a − , a, , . . . , x a − = ( a, , . . . , , b, b − a, a − )so that ( x ) = [ x a − , x a ]. Moreover for λ ∈ N , ( x ) λ ≡ [ x a − , ( x a ) λ ] mod (cid:2)(cid:2) K, K (cid:3) K (cid:3) ≤ L ( × p )2 ≤ K ( × p ) (by Lemma 4.3).Now, any element of K ( × p ) /K ( × p ) is represented by a vector (cid:0) ( x ) λ j (cid:1) pj =1 for some ( λ j ) pj =1 ∈ F pp . By the above, (cid:0) ( x ) λ j (cid:1) pj =1 ≡ (cid:2) ( x a − ) pj =1 , (( x a ) λ j ) pj =1 (cid:3) mod K ( × p ) and we are done.Now suppose p = 5. We have x a = ( b, b − a, a − , a, x , x a ] = ( x , aba − b − a, , , aba − b − a ≡ x mod L . So: x , x a ] ≡ ( x ) mod K ( × .As before, any element of K ( × ) /K ( × ) is represented by: (cid:0) ( x ) λ j (cid:1) j =1 ≡ (cid:2) ( x ) j =1 , (( x a ) λ j ) j =1 (cid:3) mod K ( × ) for some ( λ j ) j =1 ∈ F , as required.Finally suppose p = 3. Recall that: b = ( a, a − , b ) and x = [ a, b ] = ( b − a, a, ab ).Thus: [ b, x ] = ([ a, b − a ] , , [ b, ab ])We compute:[ a, b − a ] = ([ b, a ] b − ) a − ≡ ( x − ) a − mod K ( × (by Lemma 4.3 (i)) ≡ x − mod L and: [ b, ab ] = [ b, a ][[ b, a ] , b ] ≡ x − mod K ( × (by Lemma 4.3 (i))so that: [ b, x ] ≡ ( x − , , x − ) mod L ( × ≡ ( x ) mod K ( × . (16)Now: x = [ a, x ] = ( b − a − b − a, a − ba, b )= ( bx [ x , b − ] , bx − , b ) ≡ ( bx , bx − , b ) mod K ( × ) (by Lemma 4.3 (i))so for λ , λ , λ ∈ N ,[ x , ( x λ , x λ , x λ )] ≡ (cid:0) [ bx , x λ ] , [ bx − , x λ ] , [ b, x λ ] (cid:1) mod K ( × ) (by the well-definedness of c ) whereas for µ, λ ∈ N ,[ bx µ , x λ ] = [ b, x λ ][[ b, x λ ] , x µ ] ≡ [ b, x λ ] mod L ( × ≡ [ b, x ] λ mod L ( × (by Lemma 4.3 (ii))so by (16) we have: x , ( x λ , x λ , x λ )] ≡ ( ( x ) λ , ( x ) λ , ( x ) λ ) mod L ( × )2 and every element of K ( × p ) /K ( × p ) is represented by ( x λ , x λ , x λ ) for some λ i ∈ F , as required.Second, we construct an approximation to elements of K ( × p ) i + up to an errorlying in K ( × p ) i + . Proposition 4.5.
Let ≤ i ≤ p − . The restriction of c to L p − × K ( × p ) i descends to a well-defined map: ¯ c i : ( L p − /K ( × p ) ) × ( K ( × p ) i /K ( × p ) i + ) → Γ /K ( × p ) i + whose image contains K ( × p ) i + /K ( × p ) i + .Proof. We check first that ¯ c i is well-defined. Observe that by Lemma 2.1, for g ∈ L p − , h ∈ K ( × p ) i , g ∈ K ( × p ) and h ∈ K ( × p ) i + , we have:[ g g , h h ][ g , h ] − ∈ (cid:2) L p − , K ( × p ) i + (cid:3)(cid:2) K ( × p ) i , K ( × p ) (cid:3) .It therefore suffices to check that: (cid:2) L p − , K ( × p ) i + (cid:3)(cid:2) K ( × p ) i , K ( × p ) (cid:3) ⊆ K ( × p ) i + . (17)But L p − = h x p − , L p − i and L p − ⊆ Stab Γ (2), so that by Lemma 4.3 (i), (cid:2) L p − , K ( × p ) i + (cid:3) ⊆ K ( × p ) ⊆ K ( × p ) i + while x p − ( a, . . . , a ) − ∈ Stab Γ (1) ( × p ) , so: (cid:2) h x p − i , K ( × p ) i + (cid:3) ⊆ (cid:2) h a i , K ( × p ) i + (cid:3) ( × p ) (cid:2) Stab Γ (1) , K ( × p ) i + (cid:3) ( × p ) ⊆ K ( × p ) i + (using Lemma 4.3 (ii)). Thus (cid:2) L p − , K ( × p ) i + (cid:3) ⊆ K ( × p ) i + . Also, K ( × p ) i , K ( × p ) ⊆ Stab Γ (3), so by Lemma 4.3 (i), (cid:2) K ( × p ) i , K ( × p ) (cid:3) ⊆ K ( × p ) ⊆ K ( × p ) i + .Thus (17) is indeed satisfied, and ¯ c i is well-defined.We now check that the image of ¯ c i contains K ( × p ) i + /K ( × p ) i + . First note thatfor any λ ∈ N , ( i + )( x ) λ = ( i + )( x λ )= [ a, i ( x λ )]= [ a, i ( x ) λ ].Now, every element of K ( × p ) i + /K ( × p ) i + is represented by an element: (cid:0) ( i + )( x ) λ j (cid:1) pj =1 = (cid:2) ( a ) pj =1 , ( i ( x ) λ j ) pj =1 (cid:3) (18)for some ( λ j ) pj =1 ∈ N p . From (14), there exist y , . . . , y p ∈ Stab Γ (1) such that: p − = ( ay j ) pj =1 .For 1 ≤ j ≤ p ,[ ay j , i ( x ) λ j ][ a, i ( x ) λ j ] − ∈ (cid:2) Stab Γ (1) , K ( × p ) i (cid:3) ⊆ K ( × p ) i + by Lemma 2.1 and Lemma 4.3 (ii). Combining with (18) we have: (cid:0) ( i + )( x ) λ j (cid:1) pj =1 ≡ c (cid:0) x p − , ( i ( x ) λ j ) pj =1 (cid:1) mod K ( × p ) i + .Since x p − ∈ L p − and ( i ( x ) λ j ) pj =1 ∈ K ( × p ) i we are done.Finally, we construct an approximation to elements of L ( × p ) i +1 up to an errorlying in L ( × p ) i +2 . Proposition 4.6.
Let ≤ i ≤ p − . The restriction of c to L ( × p ) p − × L ( × p ) i descends to a well-defined map: ¯¯ c i : ( L ( × p ) p − /K ( × p ) ) × ( L ( × p ) i /L ( × p ) i +1 ) → Γ /L ( × p ) i +2 whose image contains L ( × p ) i +1 /L ( × p ) i +2 .Proof. We check first that ¯¯ c i is well-defined. Observe that by Lemma 2.1, for g ∈ L ( × p ) p − , h ∈ L ( × p ) i , g ∈ K ( × p ) and h ∈ L ( × p ) i +1 , we have:[ g g , h h ][ g , h ] − ∈ (cid:2) L ( × p ) p − , L ( × p ) i +1 (cid:3)(cid:2) K ( × p ) , L ( × p ) i (cid:3) .It therefore suffices to check that: (cid:2) L ( × p ) p − , L ( × p ) i +1 (cid:3)(cid:2) K ( × p ) , L ( × p ) i (cid:3) ⊆ L ( × p ) i +2 . (19)Certainly, (cid:2) K ( × p ) , L ( × p ) i (cid:3) ≤ K ( × p ) ≤ L ( × p ) i +2 (by Lemma 4.3 (i)). Meanwhile, (cid:2) L ( × p ) p − , L ( × p ) i +1 (cid:3) ⊆ (cid:2) L p − , L ( × p ) i +1 (cid:3) ( × p ) so it suffices to check that (cid:2) L p − , L ( × p ) i +1 (cid:3) ⊆ L ( × p ) i +2 . L p − is generated by x p − and L p − ⊆ Stab Γ (1) ( × p ) (the latter inclusionholds because L p − is generated by K ( × p ) and x p − ∈ ( b, . . . , b ) K ( × p ) ) Thus: (cid:2) L p − , L ( × p ) i +1 (cid:3) ⊆ (cid:2) Stab Γ (1) , L i +1 (cid:3) ( × p ) ≤ K ( × p ) ≤ L ( × p ) i +1 .Meanwhile, L ( × p ) i +1 /L ( × p ) i +2 is generated by the images of ( x i +1 ) , . . . , ( p − )( x i +1 ),so by (14), (cid:2) h x p − i , L ( × p ) i +1 (cid:3) ⊆ (cid:2) h a i , h x i +1 i (cid:3) ( × p ) L ( × p ) i +2 ⊆ L ( × p ) i +2 .Thus (19) is indeed satisfied, and ¯¯ c i is well-defined.We now check that the image of ¯¯ c i contains L ( × p ) i +1 /L ( × p ) i +2 .First, for any λ ∈ N , λi +1 = [ a, x i ] λ ≡ [ a, x λi ] mod K ( × p ) (by Lemma 4.3 (i)).As in the proof of Proposition 4.5, there exist, by (14), y , . . . , y p ∈ Stab Γ (1)such that: x p − = ( ay j ) pj =1 .For 1 ≤ j ≤ p , [ ay j , x λ j i ][ a, x λ j i ] − ∈ (cid:2) K, K (cid:3) ≤ K ( × p ) (by Lemma 4.3 (i)). Thus for λ , . . . , λ p ∈ N ,[ x p − , ( x λ i , . . . , x λ p i )] ≡ ( x λ i +1 , . . . , x λ p i +1 ) mod K ( × p ) .Now every element of L ( × p ) i +1 /L ( × p ) i +2 is represented by:( x λ i +1 , . . . , x λ p i +1 )for some ( λ j ) p j =1 ∈ N p . From the above,( x λ i +1 , . . . , x λ p i +1 ) ≡ (cid:2) ( x p − ) pj =1 , ( x λ j i ) p j =1 (cid:3) mod K ( × p ) ≤ L ( × p ) i +2 and we have ( x p − ) pj =1 ∈ L ( × p ) p − , ( x λ j i ) p j =1 ∈ L ( × p ) i so the result follows.Up to now, we have concentrated on approximating, by commutators, ele-ments lying in K ( × p ) but outside K ( × p ) . We can however quickly extend theseapproximations to elements lying between K ( × p m ) amd K ( × p m +1 ) for arbitrary m ≥
2. Indeed, since the conclusions of Propositions 4.4-4.6 concern only com-putations within the group
K/K ( × p ) the generalisation from the case m = 2 isimmediate from the identification: K ( × p m − ) /K ( × p m +1 ) ∼ = ( K/K ( × p ) ) ( × p m − ) and the observation that, in a direct product of groups, a tuple of commutatorsof elements of the factors is the commutator of the tuples of those same elements. Proposition 4.7.
Let m ≥ .(i) The restriction of c to L ( × p m − ) p − × K ( × p m − ) descends to a well-definedmap: ¯ c ,m : ( L ( × p m − ) p − /K ( × p m ) ) × ( K ( × p m − ) /K ( × p m ) ) → Γ /K ( × p m ) whose image contains K ( × p m ) /K ( × p m ) .(ii) Let ≤ i ≤ p − . The restriction of c to L ( × p m − ) p − × K ( × p m ) i descends toa well-defined map: ¯ c i,m : ( L ( × p m − ) p − /K ( × p m ) ) × ( K ( × p m ) i /K ( × p m ) i + ) → Γ /K ( × p m ) i + hose image contains K ( × p m ) i + /K ( × p m ) i + .(iii) Let ≤ i ≤ p − . The restriction of c to L ( × p m − ) p − × L ( × p m − ) i descendsto a well-defined map: ¯¯ c i,m : ( L ( × p m − ) p − /K ( × p m ) ) × ( L ( × p m ) i /L ( × p m ) i +1 ) → Γ /L ( × p m ) i +2 whose image contains L ( × p m ) i +1 /L ( × p m ) i +2 .Proof. By the preceding discussion, (i), (ii) and (iii) follow, respectively, fromPropositions 4.4, 4.5 and 4.6.We are now ready to put everything together, and use the approximationsfrom Proposition 4.7 (i)-(iii) to prove a result closely analogous to Proposition3.8: namely, if a symmetric subset X ⊆ Γ contains an approximation to ev-ery element of Γ up to an error lying in K ( × p m ) , then every element of Γ isapproximated up to an error lying in K ( × p m +1 ) by a short word in X . Proposition 4.8.
Let C p be as in Theorem 1.3. Let m ≥ and let X ⊆ Γ bea symmetric subset such that: XK ( × p m ) = Γ . (20) Then: X C p K ( × p m +1 ) = Γ . (21) Proof.
The first step shall be to show that: K ( × p m ) ⊆ X K ( × p m ) . (22)To this end let k ∈ K ( × p m ) . By Proposition 4.7 (i), there exist g ∈ L ( × p m − ) p − , h ∈ K ( × p m − ) such that: [ g, h ] ≡ k mod K ( × p m ) .From (20), there exist x g , x h ∈ X such that: x g ≡ g, x h ≡ h mod K ( × p m ) so by the well-definedness of the map ¯ c ,m from Proposition 4.7 (i), X ∋ [ x g , x h ] ≡ k mod K ( × p m ) and we have (22).Define the integer sequence ( a n ) n recursively by a = 4 and a n = 2 a n − + 2for n ≥
1. The second step of the proof shall be to show that: K ( × p m ) i ⊆ X a i K ( × p m ) i + (23)for 0 ≤ i ≤ p −
1. This shall be achieved by induction on i , using Proposition4.7 (ii) at each stage (and the base case i = 0 being provided by (22)).For let 1 ≤ i ≤ p − k ∈ K ( × p m ) i . By Proposition 4.7 (ii), there exist g ∈ L ( × p m − ) p − , h ∈ K ( × p m ) i − such that: g, h ] ≡ k mod K ( × p m ) i + .From (20) and the induction hypothesis, there exist x g ∈ X , x h ∈ X a i − suchthat: x g ≡ g mod K ( × p m ) , x h ≡ h mod K ( × p m ) i so by the well-definedness of the map ¯ c i,m from Proposition 4.7 (ii), X a i = X a i − +2 ∋ [ x g , x h ] ≡ k mod K ( × p m ) i + and we have (23).Define the integer sequence ( b n ) n recursively by b = P p − n =0 a n and b n +1 = 2 b n + 2 for n ≥
2. Combining the inclusions (23) for i from 0 to p −
1, we have: K ( × p m ) = K ( × p m ) ⊆ X b K ( × p m ) p = X b L ( × p m )2 . (24)Our third objective shall be to show that: L ( × p m ) i ⊆ X b i L ( × p m ) i +1 (25)for 1 ≤ i ≤ p −
1. This again shall be by induction on i , using Proposition 4.7(iii), the base case i = 1 being provided by (24).Thus let 2 ≤ i ≤ p − k ∈ L ( × p m ) i . By Proposition 4.7 (ii), thereexist g ∈ L ( × p m − ) p − , h ∈ L ( × p m ) i − such that:[ g, h ] ≡ k mod L ( × p m ) i +1 .By (20) and the induction hypothesis, there exist x g ∈ X , x h ∈ X b i − such that: x g ≡ g mod K ( × p m ) , x h ≡ h mod L ( × p m ) i so by the well-definedness of the map ¯¯ c i,m from Proposition 4.7 (iii), X b i = X b i − +2 ∋ [ x g , x h ] ≡ k mod L ( × p m ) i +1 as desired.Finally, set C p = 1 + P p − i =1 b i . Expressing a n and b n in closed form, C p is asin the statement of Theorems 1.3 and 1.4. We combine the inclusions (25) for i from 1 to p − K ( × p m ) = L ( × p m )1 ⊆ X C p − L ( × p m ) p = X C p − K ( × p m +1 ) .Combining this last inclusion with (20), we have Γ = X C p K ( × p m +1 ) , as required. Proof of Theorem 1.4.
Let S ⊆ Γ /K ( × p n ) . If n ≤ /K ( × p n ) ) ≤ | G : K ( × p ) | = p p +1 .f n ≥ S ⊆ Γ be any subset whose image in Γ /K ( × p n ) is S . Then B ˜ S ( p p +1 ) K ( × p ) = Γ, and by repeated application of Proposition 4.8, B ˜ S ( p p +1 C n − p ) K ( × p n ) = Γ.Thus: diam(Γ /K ( × p n ) , S ) ≤ p p +1 C n − p ≪ p p log( C p ) n/ log( p ) The result follows, since | Γ : K ( × p n ) | = p p n +1 . Proof of Theorem 1.3.
By Lemma 2.12, we have:diam(Γ / Stab Γ ( n )) ≤ diam(Γ /K ( × p n − ) ).The result now follows from | Γ : K ( × p n ) | = p p n +1 ; Theorem 1.4 and Corollary4.1. -Group In this Section we deduce Theorem 1.5 from Theorem 1.4. We shall requiresome facts about the lower central series of Γ = Γ (3) , which were established in[3]. Define the integer sequence ( α n ) by α = 1, α = 2, α n = 2 α n − + α n − for n ≥
3, and set β n = P ni =1 α i . We have: α n = 12 √ (cid:0) (1 + √ n − (1 − √ n (cid:1) ,β n = 14 (cid:0) (1 + √ n +1 + (1 − √ n +1 − (cid:1) . Theorem 5.1 ([3]) . Let X , . . . , X n ∈ { , , } . Then: deg X · · · X n ( x ) = 1 + n X i =1 X i α i + α n +1 deg X · · · X n ( x ) = 1 + n X i =1 X i α i + 2 α n +1 . Corollary 5.2.
For all m ∈ N , K ( × m ) ≤ γ α m +1 +1 (Γ) ≤ γ β m +1 (Γ) . Corollary 5.3.
For all m ∈ N , | Γ : γ β m +1 (Γ) | = 3 (3 m +1) / .roof of Theorem 1.5. For n = 1 there is nothing to prove. Otherwise, let m ∈ N be such that β m + 1 ≤ n ≤ β m +1 + 1. Then by Corollary 5.2,diam(Γ /γ n (Γ)) ≤ diam(Γ /K ( × m +1 ) ) ≪ log(111) m/ log(3) (by Theorem 1.4) ≪ n log(111) / log(1+ √ ≪ (log | Γ : γ β m +1 (Γ) | ) log(111) / log(3) (by Corollary 5.3) ≤ (log | Γ : γ n (Γ) | ) log(111) / log(3) . For G a finite group and S ⊆ G a symmetric subset, let A S be the (normalized)adjacency operator on the Cayley graph Cay( G, S ). A S is a self-adjoint operatorof norm 1; let its spectrum be:1 = λ ≥ λ ≥ . . . ≥ λ | G | ≥ − λ = 1 corresponding to the constant functionals on G .More generally, the 1-eigenspace of A S is spanned by the indicator functions ofthe connected components of Cay( G, S ); in particular 1 > λ if and only if S generates G , and in this case the quantity 1 − λ is known as the spectral gap of the pair ( G, S ).The existence of a large spectral gap for a family of Cayley graphs is a matterof great interest. If a family of finite graphs of bounded valence with vertex setsof unbounded size possess a spectral gap bounded below by an absolute positiveconstant, then the graphs form an expander family . Expander graphs (andespecially expander
Cayley graphs) have multifarious applications across puremathematics and theoretical computer science [25].Now let Γ be G or Γ ( p ) and let (Γ i ) i be one of the descending sequencesof finite-index normal subgroups from Theorems 1.1-1.5. Cayley graphs of thequotient groups Γ / Γ i do not in general form expander families: for instance if S is a finite symmetric generating set for Γ, and S i is the image of S in Γ / Γ i , thenthe spectral gap of Cay(Γ / Γ i , S i ) tends to 0 as i → ∞ (this follows from thefact that Γ is amenable [7] and Γ i exhausts Γ). We do however have a weakerlower bound on the spectral gap of any connected Cayley graph of Γ / Γ i , comingfrom our upper bounds on diameter and the following general inequality. Proposition 6.1 ([11] Corollary 3.1) . Let G be a finite group and let S be a sym-metric generating set. Then the spectral gap of ( G, S ) is ≥ ( | S | diam( G, S ) ) − . Theorems 1.1-1.5 combine with Proposition 6.1 to yield the following boundson spectral gaps. orollary 6.2.
Let S be an arbitrary generating set for the finite group G .Denote by ǫ ( G, S ) the spectral gap of the pair ( G, S ) . Let C p be as in Theorem1.3.(i) If G = G / Stab G ( n ) then ǫ ( G, S ) = Ω (cid:0) | S | − exp( − n ) (cid:1) ;(ii) If G = G /γ n ( G ) then ǫ ( G, S ) = Ω (cid:0) | S | − n − / log(2) (cid:1) ;(iii) If G = Γ ( p ) / Stab Γ ( p ) ( n ) then ǫ ( G, S ) = Ω p (cid:0) | S | − exp( − C p ) n ) (cid:1) ;(iv) If G = Γ ( p ) /K ( × p n ) then ǫ ( G, S ) = Ω p (cid:0) | S | − exp( − C p ) n ) (cid:1) ;(v) If G = Γ (3) /γ n (Γ (3) ) then ǫ ( G, S ) = Ω (cid:0) | S | − n − / log(1+ √ (cid:1) . A second closely related numerical invariant of finite Cayley graphs is the mixing time . This is a measure of the time taken for a lazy random walk onthe Cayley graph to approach the uniform distribution. It may be defined asfollows. Let f = δ e be the Dirac mass at the identity of G . The lazy randomwalk on Cay( G, S ) is defined by the operator T S = ( A S + I ) /
2, where I isthe identity operator on Cay( G, S ), and describes the progress on Cay(
G, S )of a particle which starts at the identity, and which at each step with equalprobability either traverses an edge (chosen uniformly at random) or remainsstationary. Recursively define f l +1 = T S ( f l ), the distribution of the walk attime l . We may consider the walk to be well-mixed when f l is close to theuniform distribution, in some appropriate norm on the complex functionals on G . Here we focus on mixing with respect to the ℓ ∞ -norm. Definition 6.3.
Let G be a finite group and S be a symmetric generating set.The ℓ ∞ -mixing time of the pair ( G, S ) is the smallest positive integer l suchthat: (cid:13)(cid:13) f l − | G | χ G (cid:13)(cid:13) ∞ ≤ | G | . It may be easily seen that the LHS of the above inequality is a non-increasingfunction of l , so that once the random walk reaches its mixing time, it remainswell-mixed thereafter. There is a close relationship between mixing time andspectral gap. Proposition 6.4 ([24] Theorem 5.1) . Suppose the pair ( G, S ) has spectral gap ǫ > . Then there exists an absolute constant C > such that the ℓ ∞ -mixingtime of ( G, S ) is at most ( C/ǫ ) log | G | . Applying Proposition to the conclusions of Corollary 6.2, we have corre-sponding bounds on mixing times, as follows.
Corollary 6.5.
Denote by µ ( G, S ) the ℓ ∞ -mixing time of the pair ( G, S ) .(i) If G = G / Stab G ( n ) then µ ( G, S ) = O (cid:0) | S | exp(log(2450) n ) (cid:1) ;(ii) If G = G /γ n ( G ) then µ ( G, S ) = O (cid:0) | S | n / log(2)+1 (cid:1) ;iii) If G = Γ ( p ) / Stab Γ ( p ) ( n ) then µ ( G, S ) = O p (cid:0) | S | exp(log( pC p ) n ) (cid:1) ;(iv) If G = Γ ( p ) /K ( × p n ) then µ ( G, S ) = O p (cid:0) | S | exp(log( pC p ) n ) (cid:1) ;(v) If G = Γ (3) /γ n (Γ (3) ) then µ ( G, S ) = O (cid:0) | S | n log(36963) / log(1+ √ (cid:1) . Given a finitely generated group G and a finite generating set S ⊆ G , let f ( G,S ) ( n ) = | B S ( n ) | be the growth function . Although for a given group G , thefunction f ( G,S ) may vary according to the generating set S , it only does so upto an appropriate notion of equivalence of functions. As such, we may speakwithout ambiguity about groups of polynomial growth , exponential growth andso on (see [10] Chapters VI-VII).One of the key sources of interest in branch groups is the fact that theyinclude many examples of groups with exotic growth behaviour. In particular, G has intermediate growth , that is: growth faster than any polynomial functionbut slower than any exponential function.The following elementary fact exhibits a relationship between growth anddiameter. Lemma 7.1.
Let F be a finite group, and let φ : G → F be an epimorphism.Then: f ( G,S ) (diam( F, φ ( S ))) ≥ | F | . (26)This inequality suggests the following definition, which is made by analogywith that of the diameter of a finite group. Let f G ( n ) be the minimal valueof f ( G,S ) ( n ), as S ranges over all finite generating subsets of G . From (26) weimmediately obtain: f G (diam( F )) ≥ | F | . (27)The relationship between growth and diameter can be exploited to yield in-formation about both. For instance, from Theorems 1.2 and 1.4 we have thefollowing. Corollary 7.2.
There exist constants α ( p ) > such that: f G ( n ) ≫ exp( α (2) n β ( G ) ) and f Γ ( p ) ( n ) ≫ p exp( α ( p ) n β (Γ ( p ) ) ) ,where β ( G ) = log(2) / log(35) ≈ . and β (Γ ( p ) ) = log( p ) / log( C ( p )) (here C ( p ) is as in Theorem 1.3). The bounds in Corollary 7.2 are not the best known: a slight modifica-tion of an argument of Grigorchuk [17] shows that if G is any finitely gener-ated residually virtually nilpotent group, then either G is virtually nilpotent or f G ( n ) ≥ exp( √ n ) (see [4]). In particular the latter conclusion applies to G andΓ ( p ) . It is however possible that improvements upon the diameter bounds inTheorems 1.2 and 1.4 and their corollaries could yield new lower bounds on f G and f Γ ( p ) .onversely, known upper bounds on the growth translate into lower boundson the diameters of finite quotients. In the case of G , the best upper bound onthe growth is the following result of Bartholdi. Theorem 7.3 ([16]) . Let a, b, c, d ∈ G be as in subsection 2.2.1 and let S = { a, b, c, d } . Then: f ( G ,S ) ( n ) ≪ exp( n β ) ,where β = log(2)log(2) − log( η ) ≈ . , for η the real root of X + X + X = 2 . Corollary 7.4.
There exist absolute constants
C, C ′ > such that: diam( G / Stab G ( n )) ≥ C (log | G : Stab G ( n ) | ) /β ; diam( G /γ n ( G )) ≥ C ′ (log | G : γ n ( G ) | ) /β where β is as in Theorem 7.3. Corollary 7.4 places a limit on the extent to which the constant log(35) / log(2)appearing in Theorems 1.1 and 1.2 might be reduced (though it is almost cer-tainly not sharp). It is unclear at this time whether the constant 1 /β ≈ . Acknowledgements
I would like to thank Laurent Bartholdi and Alejandra Garrido for illuminatingdiscussions. Parts of this work were completed while the author was a fellow-commoner at Trinity Hall, Cambridge. I would like to thank the fellows of theCollege for providing me with a warm welcome and pleasant working conditions.
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