PPRODUCT SET GROWTH IN BURNSIDE GROUPS
R ´EMI COULON AND MARKUS STEENBOCKA
BSTRACT . Given a periodic quotient of a torsion-free hyperbolic group, we provide a fine lower estimateof the growth function of any sub-semi-group. This generalizes results of Razborov and Safin for free groups.
1. I
NTRODUCTION If V is a subset in a group G , we denote by V r ⊂ G the set of all group elements that are representedby a product of exactly r elements of V . In this paper we are interested in the growth of V r . Such aproblem has a long history which goes back (at least) to the study of additive combinatorics. See forinstance [Nat96, TV06]. In the context of non-abelian groups, it yields to the theory of approximatesubgroups, see [Tao08, BGT12], and relates to spectral gaps in linear groups, see [Hel08, BG08, BG12], aswell as exponential growth rates of negatively curved groups, [Kou98, AL06, BF18, FS20].If G is a free group, Safin [Saf11], improving former results by Chang [Cha08] and Razborov [Raz14],proves that there exists c > such that for every finite subset V ⊂ G , either V is contained in a cyclicsubgroup, or for every r ∈ N , we have | V r | (cid:62) ( c | V | ) [( r +1) / . This estimate can be thought of as a quantified version of the Tits alternative in G . A similar statement holdsfor SL ( Z ) [Cha08], free products, limit groups [But13] and groups acting on δ -hyperbolic spaces [DS20].All these groups display strong features of negative curvature, inherited from a non-elementary acylindricalaction on a hyperbolic space. Some results are also available for solvable groups, see [Tao10, But13].By contrast, in this work, we focus on a class of groups which do not admit any non-degenerate actionon a hyperbolic space, namely the set of infinite groups with finite exponent, often referred to as ofBurnside groups.1.1. Burnside groups of odd exponent.
Given a group G and an integer n , we denote by G n thesubgroup of G generated by all its n -th powers. We are interested in quotients of the form G/G n whichwe call Burnside groups of exponent n . If G = F k is the free group of rank k , then B k ( n ) = G/G n is Date : February 23, 2021.2020
Mathematics Subject Classification.
Key words and phrases. product sets, growth, hyperbolic groups, acylindrical actions, small cancellation, infinite periodic groups,Burnside problem. a r X i v : . [ m a t h . G R ] F e b R´EMI COULON AND MARKUS STEENBOCK the free Burnside group of rank k and exponent n . The famous Burnside problem asks whether a finitelygenerated free Burnside group is necessarily finite.Here, we focus on the case that the exponent n is odd. By Novikov’s and Adian’s solution of theBurnside problem, it is known that B k ( n ) is infinite provided k (cid:62) and n is a sufficiently large oddinteger [Adi79]. See also [Ol’82, DG08]. More generally, if G is a non-cyclic, torsion-free, hyperbolicgroup, then the quotient G/G n is infinite provided n is a sufficiently large odd exponent [Ol’91, DG08].Our main theorem extends Safin’s result to this class of Burnside groups. Theorem 1.1.
Let G be a non-cyclic, torsion-free hyperbolic group. There are numbers n > and c > such that for all odd integer n (cid:62) n the following holds. Given a finite subset V ⊂ G/G n , either V iscontained in a finite cyclic subgroup, or for all r ∈ N , we have | V r | (cid:62) ( c | V | ) [( r +1) / . Observe that the constant c only depends on G and not on the exponent n . Recall that Burnside groupsdo not act, at least in any useful way, on a hyperbolic space. Indeed, any such action is either ellipticor parabolic. On the other hand, it is well-known that any linear representation of a finitely generatedBurnside group has finite image. Thus our main theorem is not a direct application of previously knownresults.Let us mention some consequences of Theorem 1.1. If V is a finite subset of a group G , one defines its entropy by h ( V ) = lim sup r →∞ r log | V r | . The group G has uniform exponential growth if there exists ε > such that for every finite symmetricgenerating subset V of G , h ( V ) > ε . In addition, G has uniform uniform exponential growth if thereexists ε > such that for every finite symmetric subset V ⊂ G , either V generates a virtually nilpotentgroup, or h ( V ) > ε . Corollary 1.2.
Let G be a non-cyclic, torsion-free hyperbolic group. There are numbers n > and α > such that for all odd integer n (cid:62) n , the following holds. Given a finite subset V ⊂ G/G n containing the identity, either V is contained in a finite cyclic subgroup, or h ( V ) (cid:62) α ln | V | (cid:62) α ln 2 . In particular,
G/G n has uniform uniform exponential growth. It was already known that free Burnside groups of sufficiently large odd exponent have uniformexponential growth, see Osin [Osi07, Corollary 1.4] and Atabekyan [Ata09, Corollary 3]. Note thatTheorem 2.7 in [Osi07] actually shows that free Burnside groups have uniform uniform exponentialgrowth. Nevertheless, to the best of our knowledge, the result was not proved for Burnside quotients
RODUCT SET GROWTH IN BURNSIDE GROUPS 3 of hyperbolic groups. We shall also stress the fact that, unlike in Corollary 1.2, the growth estimatesprovided in [Osi07, Ata09] depend on the exponent n . The reason is that the parameter M given forinstance by [Osi07, Theorem 2.7] is a quadratic function of n .Given a group G with uniform exponential growth, a natural question is whether or not there existsa finite generating set that realizes the minimal growth rate. The first inequality is a statement `a la Arzhantseva-Lysenok for torsion groups, see [AL06, Theorem 1]. The philosophy is the following: if theset V has a small entropy, then it cannot have a large cardinality. In particular, if we expect the minimalgrowth rate to be achieved, we can restrict our investigation to generating sets with fixed cardinality.Note that this is exactly the starting point of the work of Fujiwara and Sela in the context of hyperbolicgroups, [FS20].Let us discuss now the power arising in Theorem 1.1. We claim that, as our estimate is independent ofthe exponent n , the power ( r + 1) / is optimal. For this purpose we adapt an example of [Saf11]. Example 1.3.
Let g and h be two elements in B ( n ) such that g generates a group of order n , that doesnot contain h . Consider the set V N = (cid:8) , g, g , . . . , g N , h (cid:9) . Whenever the exponent n is sufficiently large compared to N , we have | V rN | ∼ N [( r +1) / while | V N | = N + 1 .Button observed the following fact. Assume that there is c > and ε > with the following property:for all finite subsets V in a group G that are not contained in a virtually nilpotent subgroup, we have | V | (cid:62) c | V | ε . Then G is either virtually nilpotent, or of bounded exponent [But13, Proposition 4.1].We do not know if such a non-virtually nilpotent group exists. Burnside groups of large exponents wouldhave been possible candidates, but by Example 1.3 this possibility can be excluded.1.2. Groups acting on hyperbolic spaces.
In the first part of our paper, we revisit product set growth fora group G acting on a hyperbolic space X , see [DS20, Theorem 1.14]. For this purpose, we use the notionof an acylindrical action, see [Sel97, Bow08]. Given a subset U of G , we exploit its (cid:96) ∞ -energy defined as λ ( U ) = inf x ∈ X sup u ∈ U | ux − x | . Theorem 1.4 (see Theorem 8.1) . Let G be group acting acylindrically on a hyperbolic length space X .There exists a constant C > such that for every finite U ⊂ G with λ ( U ) > C ,(1) either | U | (cid:54) C ,(2) or there is subset S ⊂ U freely generating a free sub-semigroup of cardinality | S | (cid:62) C λ ( U ) | U | . R´EMI COULON AND MARKUS STEENBOCK
Remark 1.5.
For simplicity we stated here a weakened form of Theorem 8.1. Actually we prove that theconstant C only depends on the hyperbolicity constant of the space X and the acylindricity parameters ofthe action of G . The set S is also what we called strongly reduced , see Definition 3.1. Roughly speakingthis means that the orbit map from the free semi-group S ∗ to X is a quasi-isometric embedding.There is quite some litterature on finding free sub-semigroups in powers of symmetric subsets U ingroups of negative curvature, see [Kou98, AL06, BF18]. We can for example compare Theorem 8.1 toTheorem 1.13 of [BF18]. In this theorem, under the additional assumption that U is symmetric, the authorsconstruct a -element set in U r that generates a free sub-semigroup, where the exponent r does onlydepend on the doubling constant of the space. Let us highlight two important differences. First we do notassume that the set U is symmetric. In particular, we cannot build the generators of a free sub-semigroupby conjugating a given hyperbolic element. Hence the proofs require different techniques. Moreover, forour purpose, it is important that the cardinality of S grows linearly with the one of U . For the optimalityof our estimates discussed in the previous paragraph, we require that it is contained in U . The prize thatwe pay for this is the correction term of the order of the (cid:96) ∞ -energy of U .As the set S constructed in Theorem 1.4 freely generates a free sub-semigroup, we immediately obtainthe following estimate on the growth of U r . Corollary 1.6 (see Corollary 8.2) . Let G be group acting acylindrically on a hyperbolic length space X .There exists a constant C > such that for every finite U ⊂ G with λ ( U ) > C , and for all integers r (cid:62) ,we have | U r | (cid:62) (cid:18) Cλ ( U ) | U | (cid:19) [( r +1) / . As in the previous statement, the constant C actually only depends on the parameters of the action of G on X . Corollary 1.6 is a variant of [DS20, Theorem 1.14], where the correction term of the order of log | U | in this theorem is replaced by a geometric quantity, the (cid:96) ∞ -energy of U .1.3. Strategy for Burnside groups.
Let us explain the main idea behind the proof of Theorem 1.1. Forsimplicity we restrict ourselves to the case of free Burnside groups of rank . Let n be a sufficientlylarge odd exponent. Any known strategy to prove the infiniteness of B ( n ) starts in the same way. Oneproduces a sequence of groups(1) F = G → G → G → · · · → G i → G i +1 → . . . that converges to B ( n ) where each G i is a hyperbolic group obtained from G i − by means of smallcancellation. The approach provided by Delzant and Gromov associates to each group G i a hyperbolicspace X i on which it acts properly co-compactly. An important point is that the geometry of X i issomewhat “finer” than the one of the Cayley of G i . In particular, one controls uniformly along thesequence ( G i , X i ) , the hyperbolicity constant of X i as well as the acylindricity parameters of the action RODUCT SET GROWTH IN BURNSIDE GROUPS 5 of G i , see Proposition 10.1. As we stressed before the constant C involved in Theorem 1.4 only dependson those parameters. Thus it holds, with the same constant C , for each group G i acting on X i .Consider now a subset V ⊂ B ( n ) that is not contained in a finite subgroup. Our idea is to choose asuitable step j and a pre-image U j in G j such that the (cid:96) ∞ -energy λ ( U j ) is greater than C and at the sametime bounded from above by a constant C (cid:48) that does not depend on j . See the construction in (14). ByTheorem 1.4, we find a “large” subset S ⊂ U j that freely generates a free sub-semigroup. By large wemean that the cardinality of S is linearly bounded from below by the cardinality of U j (hence of V ).At this point we get an estimate for the cardinality of S r , hence for the one of U rj ⊂ G j , seeCorollary 1.6. However the map G j → G/G n is not one-to-one. Nevertheless there is a sufficientcondition to check whether two elements g and g (cid:48) in G j have distinct images in G/G n : roughly speaking,if none of them “contains a subword” of the form u m , with m (cid:62) n/ , then g and g (cid:48) have distinct imagesin G/G n . This formulation is purposely vague here. We refer the reader to Definition 4.1 for a rigorousdefinition of power-free elements in G j . In particular, the projection G j → G/G n is injective whenrestricted to a suitable set of power-free elements. Hence it suffices to count the number of power-freeelements in S r . This is the purpose of Sections 3 and 4. The computation is done by induction on r following the strategy of the first author from [Cou13].Again, we would like to draw the attention of the reader to the fact that in this procedure, we took greatcare to make sure that all the involved parameters do not depend on j .1.4. Burnside groups of even exponent.
Burnside groups of even exponent have a considerably differentalgebraic structure. For instance it turns out that the approximation groups G j in the sequence (1) containelementary subgroups of the form D ∞ × F where F is a finite subgroup with arbitrary large cardinalitythat embeds in a product of dihedral groups. In particular one cannot control acylindricity parametersalong the sequence ( G i ) , which means that our strategy fails here. It is very plausible that Burnsidegroups of large even exponents have uniform uniform exponential growth. Nevertheless we wonder ifTheorem 1.1 still holds for such groups.1.5. Acknowledgments.
The first author acknowledges support from the
Centre Henri Lebesgue
ANR-11-LABX-0020-01 and the
Agence Nationale de la Recherche under Grant
Dagger
ANR-16-CE40-0006-01.The second author was supported by the
Austrian Science Fund (FWF) project J 4270-N35 and thanksthe Universit´e de Rennes 1 for hospitality during his stay in Rennes. The second author thanks ThomasDelzant for related discussion during his stay in Strasbourg. We thank the coffeeshop
Bourbon d’Arsel forwelcoming us when the university was closed down during the pandemic, and for serving a wonderfulorange cake.
R´EMI COULON AND MARKUS STEENBOCK
2. H
YPERBOLIC GEOMETRY
We collect some facts on hyperbolic geometry in the sense of Gromov [Gro87], see also [CDP90,GdlH90].2.1.
Hyperbolic spaces.
Let X be a metric length space. The distance of two points x and y in X isdenoted by | x − y | , or | x − y | X if we want to indicate that we measure the distance in X . If A ⊂ X is a set and x a point, we write d ( x, A ) = inf a ∈ A | x − a | to denote the distance from x to A . Let A + α = { x ∈ X | d ( x, A ) (cid:54) α } be the α -neighbourhood of A . Given x, y ∈ X , we write [ x, y ] fora geodesic from x to y (provided that such a path exists). Recall that there may be multiple geodesicsjoining two points. We recall that the Gromov product of y and z at x is defined by ( y, z ) x = 12 ( | y − x | + | z − x | − | y − z | ) . Definition 2.1.
Let δ (cid:62) . The space X is δ -hyperbolic if for every x , y , z and t ∈ X ,(2) ( x, z ) t (cid:62) min { ( x, y ) t , ( y, z ) t } − δ. If δ = 0 and X is geodesic, then X is an R -tree. From now on, we assume that δ > and that X is a δ -hyperbolic metric length space.2.2. Quasi-geodesics.
A rectifiable path γ : [ a, b ] → X is a ( k, (cid:96) ) -quasi-geodesic if for all [ a (cid:48) , b (cid:48) ] ⊂ [ a, b ] the length( γ [ a (cid:48) , b (cid:48) ]) (cid:54) k | γ ( a (cid:48) ) − γ ( b (cid:48) ) | + (cid:96) ; and γ is a L -local ( k, (cid:96) ) -quasi-geodesic if any subpath of γ whose length is at most L is a ( k, (cid:96) ) -quasi-geodesic. The next lemma is used to construct (bi-infinite) quasi-geodesics. Lemma 2.2 (Discrete quasi-geodesics [Del96, Proposition 1.3.4]) . Let x , . . . , x n be n points of X .Assume that for every i ∈ { , . . . , n − } , ( x i − , x i +1 ) x i + ( x i , x i +2 ) x i +1 < | x i − x i +1 | − δ. Then the following holds(1) | x − x n | (cid:62) n − (cid:88) i =1 | x i − x i +1 | − n − (cid:88) i =2 ( x i − , x i +1 ) x i − n − δ .(2) ( x , x n ) x j (cid:54) ( x j − , x j +1 ) x j + 2 δ , for every j ∈ { , . . . , n − } .(3) If, in addition, X is geodesic, then [ x , x n ] lies in the δ -neighborhood of the broken geodesic γ = [ x , x ] ∪ · · · ∪ [ x n − , x n ] , while γ is contained in the r -neighborhood of [ x , x n ] , where r = sup (cid:54) i (cid:54) n − ( x i − , x i +1 ) x i + 14 δ. (cid:3) RODUCT SET GROWTH IN BURNSIDE GROUPS 7
We denote by L the smallest positive number larger than such that for every (cid:96) ∈ [0 , δ ] , theHausdorff distance between any two L δ -local (1 , (cid:96) ) -quasi-geodesic with the same endpoints is at most (2 (cid:96) + 5 δ ) . See [Cou14, Corollaries 2.6 and 2.7].2.3. Quasi-convex subsets.
A subset Y ⊂ X is α -quasi-convex if for all two points x, y ∈ Y , and forevery point z ∈ X , we have d ( z, Y ) (cid:54) ( x, y ) z + α . For instance, geodesics are δ -quasi-convex.If Y ⊂ X , we denote by | . | Y the length metric induced by the restriction of | . | X to Y . A subset Y that is connected by rectifiable paths is strongly-quasi-convex if it is δ -quasi-convex and if for all y, y (cid:48) ∈ Y , | y − y (cid:48) | X (cid:54) | y − y (cid:48) | Y (cid:54) | y − y (cid:48) | X + 8 δ. Isometries.
Let G be a group that acts by isometries on X .Let g ∈ G . The translation length of g is (cid:107) g (cid:107) = inf x ∈ X | gx − x | . The stable translation length of g is (cid:107) g (cid:107) ∞ = lim n →∞ n | g n x − x | . Those two quantities are related by the following inequality: (cid:107) g (cid:107) ∞ (cid:54) (cid:107) g (cid:107) (cid:54) (cid:107) g (cid:107) ∞ + 16 δ . See [CDP90,Chapitre 10, Proposition 6.4]. The isometry g is hyperbolic if, and only if, its stable translation length ispositive, [CDP90, Chapitre 10, Proposition 6.3]. Definition 2.3.
Let d > and S ⊂ G . The set of d -quasi-fixpoints of S is defined by Fix(
S, d ) = { x ∈ X | for all s ∈ S | sx − x | < d } . The axis of g ∈ G is the set A g = Fix( g, (cid:107) g (cid:107) + 8 δ ) . Lemma 2.4 ([Cou18b, Lemma 2.9]) . Let S ⊂ G . If d > δ , then the set of d -quasi-fixpoints of S is δ -quasi-convex. Moreover, assuming that Fix(
S, d ) is non-empty(1) if x ∈ X \ Fix(
S, d ) , then sup s ∈ S | sx − x | (cid:62) d ( x, Fix(
S, d )) + d − δ. (2) if L = sup s ∈ S | sx − x | − d , then x ∈ Fix(
S, d ) + L/ δ . (cid:3) Corollary 2.5 ([DG08, Proposition 2.3.3]) . Let g be an isometry of X . Then A g is δ -quasi-convex and g -invariant. Moreover, for all x ∈ X , (cid:107) g (cid:107) + 2 d ( x, A g ) − δ (cid:54) | gx − x | (cid:54) (cid:107) g (cid:107) + 2 d ( x, A g ) + 8 δ. (cid:3) R´EMI COULON AND MARKUS STEENBOCK
Acylindricity.
We recall the definition of an acylindrical action. The action of G on X is acylindrical if there exists two functions N, κ : R + → R + such that for every r (cid:62) , for all points x and y at distance | x − y | (cid:62) κ ( r ) , there are at most N ( r ) elements g ∈ G such that | x − gx | (cid:54) r and | y − gy | (cid:54) r .Recall that we assumed X to be δ -hyperbolic, with δ > . In this context, acylindricity satisfies alocal-to-global phenomenon: if there exists N , κ ∈ R + such that for all points x and y at distance | x − y | (cid:62) κ , there are at most N elements g ∈ G such that | x − gx | (cid:54) δ and | y − gy | (cid:54) δ , thenthe action of G is acylindrical, with the following estimates for the functions N and κ :(3) κ ( r ) = κ + 4 r + 100 δ and N ( r ) = (cid:16) r δ + 3 (cid:17) N . See [DGO17, Proposition 5.31]. This motivates the next definition.
Definition 2.6.
Let
N, κ ∈ R + . The action of G on X is ( N, κ ) -acylindrical if for all points x and y atdistance | x − y | (cid:62) κ , there are at most N elements g ∈ G such that | x − gx | (cid:54) δ and | y − gy | (cid:54) δ .We need the following geometric invariants of the action of G on X . The limit set of G acting on X consists of the accumulation points in the Gromov boundary ∂X of X of the orbit of one (and hence any)point in X . By definition, a subgroup E of G is elementary if the limit set of E consists of at most twopoints. Definition 2.7.
The injectivity radius is defined as τ ( G, X ) = inf {(cid:107) g (cid:107) ∞ | g ∈ G is a hyperbolic isometry } . Definition 2.8.
The acylindricity parameter is defined as A ( G, X ) = sup S diam (Fix( S, L δ )) , where S runs over the subsets of G that do not generate an elementary subgroup. Definition 2.9.
The ν -invariant is the smallest natural number ν = ν ( G, X ) such that for every g ∈ G and every hyperbolic h ∈ G the following holds: if g , hgh − , . . . , h ν gh − ν generate an elementarysubgroup, then so do g and h .The parameters A ( G, X ) and ν ( G, X ) allow us to state the following version of Margulis’ lemma. Proposition 2.10 (Proposition 3.5 of [Cou18b]) . Let S be a subset of G . If S does not generate anelementary subgroup, then, for every d > , we have Fix(
S, d ) (cid:54) [ ν ( G, X ) + 3] d + A ( G, X ) + 209 δ. (cid:3) RODUCT SET GROWTH IN BURNSIDE GROUPS 9
If there is no ambiguity we simply write τ ( G ) , A ( G ) , and ν ( G ) for τ ( G, X ) , A ( G, X ) , and ν ( G, X ) respectively. Sometimes, if the context is clear, we even write τ , A , or ν .If the action of G on X is ( N, κ ) -acylindrical, then τ (cid:62) δ/N , while A and ν are finite. In fact, onecould express upper bounds on A and ν in terms of N , κ , δ , and L . See for instance [Cou16, Section 6].However, for our purpose we need a finer control on these invariants.From now on we assume that κ (cid:62) δ and that the action of G on X is ( N, κ ) -acylindrical.2.6. Loxodromic subgroups.
An elementary subgroup is loxodromic if it contains a hyperbolic element.Equivalently, an elementary subgroup is loxodromic if it has exactly two points in its limit set. If h isa hyperbolic isometry, we denote by E ( h ) the maximal loxodromic subgroup containing h . Let E + ( h ) be the maximal subgroup of E ( h ) fixing pointwise the limit set of E ( h ) . It is know that the set F ofall elliptic elements of E + ( h ) forms a (finite) normal subgroup of E + ( h ) and the quotient E + ( h ) /F isisomorphic to Z . We say that h is primitive if its image in E + ( h ) /F generates the quotient. Definition 2.11 (Invariant cylinder) . Let E be a loxodromic subgroup with limit set { ξ, η } . The E -invariant cylinder , denoted by C E , is the δ -neighbourhood of all L δ -local (1 , δ ) -quasi-geodesics withendpoints ξ and η at infinity. Lemma 2.12 (Invariant cylinder) . Let E be a loxodromic subgroup. Then • C E is δ -quasi-convex and invariant under the action of E . If, in addition, X is proper andgeodesic, then C E is strongly quasi-convex [Cou14, Lemma 2.31], • if g ∈ E and (cid:107) g (cid:107) > L δ , then A g ⊂ C E , [Cou14, Lemma 2.33], • if g ∈ E is hyperbolic, then C E ⊂ A +52 δg . In particular, if x ∈ C E , then | gx − x | (cid:54) (cid:107) g (cid:107) + 112 δ ,[Cou14, Lemma 2.32].
3. P
ERIODIC AND APERIODIC WORDS
Let S be a finite subset of G . We denote by S ∗ the free monoid generated by S . We write π : S ∗ → G for the canonical projection. In case there is no ambiguity, we make an abuse of notations and still write w for an element in S ∗ and its image under π . We fix a base point p ∈ X . Recall that the action of G on X is ( N, κ ) -acylindrical. Definition 3.1.
Let α > . We say that the subset S is α -reduced if • ( s − p, s p ) p < α for every s , s ∈ S , • | sp − p | > α + 200 δ for every s ∈ S .The set S is α -strongly reduced if, in addition, for every distinct s , s ∈ S , we have ( s p, s p ) p < min {| s p − p | , | s p − p |} − α − δ. We say that S is reduced (respectively strongly reduced ) if there exists α > such that S is α -reduced(respectively α -strongly reduced). Lemma 3.2. If S is α -strongly reduced, then S freely generates a free sub-semi-group of G . Moreover S satisfies the geodesic extension property , that is if w, w (cid:48) ∈ S ∗ are such that ( p, w (cid:48) p ) wp (cid:54) α + 120 δ , then w is a prefix of w (cid:48) . Remark 3.3.
Roughly speaking, the geodesic extension property has the following meaning: if thegeodesic [ p, w (cid:48) p ] extends [ p, wp ] as a path in X , then w (cid:48) extends w as a word over S . Proof.
We first prove the geodesic extension property. Let w = s . . . s m and w (cid:48) = s (cid:48) . . . s (cid:48) m (cid:48) be twowords in S ∗ such that ( p, w (cid:48) p ) wp (cid:54) α + 120 δ . We denote by r the largest integer such that s i = s (cid:48) i forevery i ∈ { , . . . , r − } . For simplicity we let q = s . . . s r − p = s (cid:48) . . . s (cid:48) r − p. Assume now that contrary to our claim w is not a prefix of w (cid:48) , that is r − < m . We claim that ( wp, w (cid:48) p ) q < | s r p − p | − α − δ . If r − m (cid:48) , then w (cid:48) p = q and the claim holds. Hence we cansuppose that r − < m (cid:48) . It follows from our choice of r that s r (cid:54) = s (cid:48) r . We let t = s . . . s r p and t (cid:48) = s (cid:48) . . . s (cid:48) r p. Since S is α -strongly reduced, we have ( t, t (cid:48) ) q = ( s r p, s (cid:48) r p ) p < min {| s r p − p | , | s (cid:48) r p − p |} − α − δ. It follows then from the four point inequality that min { ( t, wp ) q , ( wp, w (cid:48) p ) q , ( w (cid:48) p, t (cid:48) ) q } (cid:54) ( t, t (cid:48) ) q + 2 δ < min {| s r p − p | , | s (cid:48) r p − p |} − α − δ. However by Lemma 2.2, we know that ( t, wp ) q = | q − t | − ( q, wp ) t (cid:62) | s r p − p | − α − δ. Thus the minimum in the previous inequality cannot be achieved by ( t, wp ) q . Similarly, it cannot beachieved by ( w (cid:48) p, t (cid:48) ) q either. Thus ( wp, w (cid:48) p ) q < min {| s r p − p | , | s (cid:48) r p − p |} − α − δ (cid:54) | s r p − p | − α − δ, which completes the proof of our claim. Using Lemma 2.2, we get ( q, w (cid:48) p ) wp (cid:62) | wp − q | − ( wp, w (cid:48) p ) q (cid:62) | s r p − p | − ( wp, w (cid:48) p ) q > α + 148 δ Applying again the four point inequality, we get α + 120 δ (cid:62) ( p, w (cid:48) p ) wp (cid:62) min { ( q, w (cid:48) p ) wp , ( p, q ) wp } − δ RODUCT SET GROWTH IN BURNSIDE GROUPS 11
It follows from our previous computation that the minimum cannot be achieved by ( q, w (cid:48) p ) wp . Usingagain Lemma 2.2 we observe that ( p, q ) wp = | wp − q | − ( p, wp ) q (cid:62) | s r p − p | − α − δ > α + 198 δ. Hence the minimum cannot be achieved by ( p, q ) wp either, which is a contradiction. Consequently w is aprefix of w (cid:48) .Let us prove now that S freely generates a free sub-semi-group of G . Let w , w ∈ S ∗ whose imagesin G coincide. In particular ( p, w p ) w p = 0 = ( p, w p ) w p . It follows from the geodesic extensionproperty that w is a prefix of w and conversely. Thus w = w as words in S ∗ . (cid:3) Periodic words.
From now on, we assume that S is α -strongly reduced (in the sense Definition 3.1).We let λ = max s ∈ S | sp − p | . We denote by | w | S the word metric of w ∈ S ∗ . Given an element w = s · · · s m in S ∗ , we let [ w ] = { p, s p, s s p, . . . , wp } . Definition 3.4.
Let E be a maximal loxodromic subgroup. We say that a word v ∈ S ∗ is m -periodic withperiod E if [ v ] ⊂ C + α +100 δE and | p − vp | (cid:62) mτ ( E ) . Remark 3.5.
Let E be a loxodromic subgroup such that p belongs to the ( α + 100 δ ) -neighborhood of C E . Let v ∈ S ∗ whose image in G is a hyperbolic element of E . Then for every integer m (cid:62) , theelement v m is m -periodic with period E . The converse is not true; that is, an m -periodic word with period E is not necessarily contained in E .If m is sufficiently large, then periods are unique in the following sense. Proposition 3.6.
There exists m (cid:62) which only depends on δ , A , ν , τ and α such that for every m (cid:62) m the following holds. If v ∈ S ∗ is m -periodic with period E and E , then E = E .Proof. Let h ∈ E realise τ ( E ) , and h ∈ E realise τ ( E ) . If v is m -periodic with period E and E ,then diam (cid:0) C + α +100 δE ∩ C + α +100 δE (cid:1) (cid:62) m max {(cid:107) h (cid:107) ∞ , (cid:107) h (cid:107) ∞ } . Recall that C E i ⊂ A +52 δh i , see Lemma 2.12. By [Cou14, Lemma 2.13] we have diam (cid:0) C + α +100 δE ∩ C + α +100 δE (cid:1) (cid:54) diam (cid:0) A +13 δh ∩ A +13 δh (cid:1) + 2 α + 308 δ. Hence there exists m (cid:62) which only depends on δ , A , ν , τ and α such that if m (cid:62) m , we have diam (cid:0) A +13 δh ∩ A +13 δh (cid:1) > ( ν + 2) max {(cid:107) h (cid:107) , (cid:107) h (cid:107)} + A + 680 δ. It follows from [Cou16, Proposition 3.44] that h and h generates an elementary subgroup, hence E = E . (cid:3) Remark 3.7.
For all w ∈ S ∗ , we have λ | w | S (cid:62) | wp − p | . In particular, if w is an m -periodic word withperiod E , then | w | S (cid:62) mτ ( E ) /λ. Conversely, by Lemma 2.2, | wp − p | (cid:62) δ | w | S . If, in addition, [ w ] ⊂ C + α +100 δE but w is not m -periodicwith period E , then | w | S (cid:54) mτ ( E ) /δ. Proposition 3.8.
Let E be a loxodromic subgroup. Let m (cid:62) be an integer. There are at most twoelements in S ∗ which are m -periodic with period E , but whose proper prefixes are not m -periodic.Proof. Let E be a maximal loxodromic subgroup. Let P E be the set of m -periodic words w ∈ S ∗ withperiod E . Observe also that if P E is non-empty, then p lies in the ( α + 100 δ ) -neighbourhood of C E .Let η − and η + be the points of ∂X fixed by E . Given any w ∈ P E , we have ( p, η − ) wp (cid:54) α + 103 δ or ( p, η + ) wp (cid:54) α + 103 δ . This yields a decomposition of P E as two subsets P − E and P + E .We are going to prove that P + E ∩ S ∗ contains at most one word satisfying the proposition. Let w and w be two words in P + E ∩ S ∗ which are m -periodic with period E , and whose proper prefixes are not m -periodic. Recall that ( p, η + ) w p (cid:54) α + 103 δ and ( p, η + ) w p (cid:54) α + 103 δ . Hence up to permuting w and w it follows from hyperbolicity that ( p, w p ) w p (cid:54) α + 104 δ . Applying the geodesic extensionproperty (see Lemma 3.2) we get that w is a prefix of w . As w is m -periodic, it cannot be a properprefix, hence w = w . Similarly, P − E ∩ S ∗ has at most one element satisfying the statement. (cid:3) The growth of aperiodic words.Definition 3.9.
Let w ∈ S ∗ and let E be a maximal loxodromic subgroup. We say that the word w contains an m -period of E if w splits as w = w w w , where the word w is m -periodic with period E .If the word w does not contain any m -period, we say that w is m -aperiodic .Observe that containing a period is a property of the word w ∈ S ∗ and not of its image π ( w ) in G : onecould find two words w and w , where w is m -aperiodic while w is not, and that have the same imagein G . However since S is strongly reduced, it freely generates a free sub-semigroup of G . Hence thispathology does not arise in our context.We denote by S ∗ m the set of m -aperiodic words in S ∗ . Recall that p is a base point of X and theparameter λ is defined by λ = max s ∈ S | sp − p | . Example 3.10. If m > λ/τ , then S ⊆ S ∗ m . Indeed, for all s ∈ S and loxodromic subgroups E , | s | S (cid:54) < mτ /λ (cid:54) mτ ( E ) /λ. So, by Remark 3.7, s cannot be m -periodic. RODUCT SET GROWTH IN BURNSIDE GROUPS 13
Recall that S r stands for the sphere of radius r in S ∗ . We denote by W ( r ) ⊂ S ∗ the ball or radius r ,that is the subset of elements w ∈ S ∗ of word length | w | S (cid:54) r . We note that | W ( r ) | (cid:54) | S | r +1 . Proposition 3.11.
Let S be a α -strongly reduced subset of G . There exists m which only depends on λ , α , A , ν , τ , and δ with the following property. For all m (cid:62) m , and r > , we have | S ∗ m ∩ W ( r + 1) | (cid:62) | S | | S ∗ m ∩ W ( r ) | . Proof.
We adapt the counting arguments of [Cou13]. We firstly fix some notations. Let m be theparameter given by Proposition 3.6. Recall that m only depends on α , A , ν , τ , and δ . Let S ⊂ G be an α -strongly reduced subset. Let m > m + 5 λ/τ . We let Z = { w ∈ S ∗ | w = w s, w ∈ S ∗ m , s ∈ S } . We denote by E the set of all maximal loxodromic subgroups in G . For each E ∈ E , let Z E ⊂ Z be thesubset of all w ∈ Z that split as a product w = w w , where w ∈ S ∗ m and w ∈ S ∗ is an m -periodicword with period E . Lemma 3.12.
The set Z \ (cid:83) E ∈E Z E is contained in S ∗ m .Proof. Let w ∈ Z contain an m -period of a loxodromic subgroup E ∈ E . By definition of Z , we have w = w s , where s ∈ S and the prefix w ∈ S ∗ does not contain any m -period. On the other hand w contains a subword w which is an m -period with period E . Since w cannot be a subword of w , it is asuffix of w . (cid:3) Recall that if U ⊂ S ∗ , then | U | stands for the cardinality of the image of U in G . However, since S freely generates a free sub-semi-group (Lemma 3.2), we can safely identify the elements of S ∗ with theirimages in G . It follows from Lemma 3.12, that for all natural numbers r , | S ∗ m ∩ W ( r ) | (cid:62) | Z ∩ W ( r ) | − (cid:88) E ∈E | Z E ∩ W ( r ) | . (4)The next step is to estimate each term in the above inequality. Lemma 3.13.
For all real numbers r , | Z ∩ W ( r + 1) | (cid:62) | S || S ∗ m ∩ W ( r ) | . Proof.
It is a direct consequence of the fact that S freely generates a free sub-semi-group. (cid:3) Lemma 3.14.
Let E ∈ E . For all real numbers r , | Z E ∩ W ( r ) | (cid:54) | S ∗ m ∩ W ( r − mτ ( E ) /λ ) | . Proof.
Let w ∈ Z E ∩ W ( r ) . By definition, w splits as a product w = w w , where w ∈ S ∗ m and w ∈ S ∗ is m -periodic with period E . By Remark 3.7, | w | S (cid:62) mτ ( E ) /λ , so that w ∈ S ∗ m ∩ W ( r − mτ ( E ) /λ ) .Since w also belongs to Z , the prefix consisting of all but the last letter does not contain m -periods.Thus every proper prefix of w cannot be m -periodic. It follows from Lemma 3.8 that there are at mosttwo possible choices for w . Hence the result. (cid:3) Lemma 3.15.
For all real numbers r , the following inequality holds: (cid:88) E ∈E | Z E ∩ W ( r ) | (cid:54) | S | m τ/δ +2 (cid:88) j (cid:62) | S ∗ m ∩ W ( r − jmτ /λ ) | | S | jm τ/δ . Proof.
Given j (cid:62) , we define E j as the set of all maximal loxodromic subgroups E ∈ E , such that jτ (cid:54) τ ( E ) < ( j + 1) τ and S ∗ contains a word that is m -periodic with period E . We split the left-handsum as follows (cid:88) E ∈E | Z E ∩ W ( r ) | = (cid:88) j (cid:62) (cid:88) E ∈E j | Z E ∩ W ( r ) | Indeed if S ∗ does not contain a word that is m -periodic with period E , then the set Z E is empty. Observethat for every E ∈ E j we have by Lemma 3.14 | Z E ∩ W ( r ) | (cid:54) | S ∗ m ∩ W ( r − jmτ /λ ) | . Thus it suffices to bound the cardinality of E j for every j (cid:62) .Let j (cid:62) . For simplicity we let d j = ( j + 1) m τ /δ + 1 . We claim that |E j | (cid:54) | S | d j +1 . To that endwe are going to build a one-to-one map from χ : E j → W ( d j ) . Indeed the cardinality of the ball W ( d j ) isat most | S | d j +1 . Let E ∈ E j . By definition there exists w ∈ S ∗ which is m -periodic with period E . Let w (cid:48) be the shortest prefix of w that is m -periodic with period E . Note that such prefix always exists since m (cid:62) m . By Remark 3.7, w (cid:48) belongs to W ( m τ ( E ) /δ + 1) hence to W ( d j ) . We define χ ( E ) to be w (cid:48) .Observe that there is at most one E such that w (cid:48) is m -periodic with period E (Proposition 3.6). Hence χ is one-to-one. This completes the proof of our claim and the lemma. (cid:3) We now complete the proof of Proposition 3.11. Let us define first some auxiliary parameters. We fixonce for all an arbitrary number (cid:15) ∈ (0 , / . In addition we let µ = (1 − (cid:15) ) | S | , γ = | S | m τ/δ , ξ = 2 | S | m τ/δ +2 , σ = (cid:15) − (cid:15) ) ξ , and M = (cid:106) mτλ (cid:107) . Since | S | (cid:62) , we observe that σ (cid:54) / . We claim that there exists an integer m (cid:62) m which onlydepends on λ , α , A , ν , τ , and δ such that γµ M (cid:54) σ, provided that m (cid:62) m . The computation shows that ln (cid:18) γσµ M (cid:19) (cid:54) (cid:18) m τδ + 3 − mτλ (cid:19) ln | S | − ln (cid:18) (cid:15) − (cid:15) ) (cid:19) − mτλ ln(1 − (cid:15) ) . RODUCT SET GROWTH IN BURNSIDE GROUPS 15
Recall that | S | (cid:62) . Hence if m (cid:62) m λδ + 3 λτ then the previous inequality yields(5) ln (cid:18) λσµ M (cid:19) (cid:54) − mτλ [ln 2 + ln(1 − (cid:15) )] + (cid:18) m τδ + 3 (cid:19) ln 2 − ln (cid:18) (cid:15) − (cid:15) ) (cid:19) . We can see from there, that there exists m (cid:62) m which only depends on λ , m , τ , and δ , such that assoon as m (cid:62) m the right hand side of Inequality (5) is non-positive, which completes the proof of ourclaim. Up to increasing the value of m , we can assume that M (cid:62) , provided m (cid:62) m .Let us now estimate the number of aperiodic words in S ∗ . From now on we assume that m (cid:62) m . Forevery integer r , we let c ( r ) = | S ∗ m ∩ W ( r ) | . We claim that for every integer r , we have c ( r ) (cid:62) µc ( r − . The proof goes by induction on r . In viewof Example 3.10, the inequality holds true for r = 1 . Assume that our claim holds for every s (cid:54) r . Inparticular for every integer t (cid:62) , we get c ( r − t ) (cid:54) µ − t c ( r ) . It follows from (4) that c ( r + 1) (cid:62) | Z ∩ W ( r + 1) | − (cid:88) E ∈E | Z E ∩ W ( r + 1) | . Applying Lemmas 3.13 and 3.15, we get c ( r + 1) (cid:62) | S | c ( r ) − ξ (cid:88) j (cid:62) c ( r + 1 − jM ) γ j . Note that jM − (cid:62) , for every j (cid:62) . Thus applying the induction hypothesis we get c ( r + 1) (cid:62) − ξµ | S | (cid:88) j (cid:62) (cid:18) γµ M (cid:19) j | S | c ( r ) . We defined µ as µ = (1 − (cid:15) ) | S | , hence it suffices to prove that ξµ | S | (cid:88) j (cid:62) (cid:18) γµ M (cid:19) j (cid:54) (cid:15). Recall that γ/µ M (cid:54) σ (cid:54) / . Hence the series converges. Moreover ξµ | S | (cid:88) j (cid:62) (cid:18) γµ M (cid:19) j (cid:54) ξµ | S | σ − σ (cid:54) ξµσ | S | (cid:54) (cid:15). This completes the proof of our claim for r + 1 . (cid:3)
4. P
OWER - FREE ELEMENTS
Let G be a group that acts ( N, κ ) -acylindrically on a δ -hyperbolic geodesic space X . We fix a basepoint p ∈ X . Definition 4.1.
Let m (cid:62) be an integer. An element g ∈ G contains an m -power if there is a maximalloxodromic subgroup E and a geodesic [ p, gp ] such that diam (cid:0) [ p, gp ] +5 δ ∩ C +5 δE (cid:1) > mτ ( E ) . If g ∈ G does not contain any m -power, we say that g is m -power-free .Let S ⊂ G be a finite subset. We recall that λ = max s ∈ S | sp − p | and that S ∗ is the set of all wordsover the alphabet S . The notions of power-free element in G and aperiod word in S ∗ are related as follows. Proposition 4.2.
Let S ⊂ G be a finite α -reduced subset. Let w ∈ S ∗ . If w contains an m -power (as anelement of G ), then w contains an m (cid:48) -period (as a word over S ), where m (cid:48) (cid:62) m − (2 λ + 20 δ ) /τ .Proof. Without loss of generality we can assume that mτ > λ + 20 δ . Indeed otherwise the statementis void. Let w = s · · · s l . As w contains a m -power, there is a loxodromic subgroup E and a geodesic [ p, wp ] such that diam (cid:0) [ p, wp ] +5 δ ∩ C +5 δE (cid:1) > mτ ( E ) . Let x , x in [ p, wp ] +5 δ ∩ C +5 δE such that | x − x | > mτ ( E ) . Let γ w = [ p, s p ] ∪ s [ p, s p ] ∪ . . . ∪ ( s · · · s l − )[ p, s l p ] be a broken geodesic joining p to wp . Let p and p be the respective projections of x and x on γ w . By Lemma 2.2, the geodesic [ p, wp ] is contained in the δ -neighborhood of γ w . Hence p and p are δ -close to C E . Moreover, | p − p | (cid:62) | x − x | − δ > mτ ( E ) − δ. Up to permuting x and x we can assume that p , p , p and wp are ordered in this way along γ w .In particular, there is i (cid:54) l − such that p ∈ ( s · · · s i ) · [ p, s i +1 p ] , and j (cid:54) l − such that p ∈ ( s · · · s j ) · [ p, s j +1 p ] . Since p comes before p on γ w , we have i (cid:54) j . Note that actually i < j . Indeedif i = j , we would have λ (cid:62) | s i p − p | (cid:62) | p − p | (cid:62) mτ ( E ) − δ (cid:62) mτ − δ, which contradicts our initial assumption. Let us set w = s · · · s i +1 and take the word w such that s · · · s j = w w . At this stage w could be the empty word. But we will see that this is not the case.Indeed | p − p | (cid:54) | p − w p | + | w p − w w p | + | w w p − p | (cid:54) | p − w p | + 2 λ. Thus, | p − w p | > mτ ( E ) − λ − δ (cid:62) m (cid:48) τ ( E ) RODUCT SET GROWTH IN BURNSIDE GROUPS 17
Applying Lemma 2.2 to the subpath γ (cid:48) of γ w bounded by p and p , we get that γ (cid:48) lies in the ( α + 14 δ ) -neighborhood of the geodesic [ p , p ] . However p and p are in the δ -neighborhood of C E which is δ -quasi-convex. Thus γ (cid:48) is contained in the ( α + 31 δ ) -neighborhood of C E . We conclude that w is m (cid:48) -periodic with period w − Ew . (cid:3)
5. E
NERGY AND QUASI - CENTER
Let G be a group acting by isometries on a δ -hyperbolic length space X . Recall that we assume forsimplicity that δ > . Let U ⊂ G be a finite subset. In order to apply the counting results from Section 3,we explain in this section and the followings how to build a strongly reduced subset S ⊂ U . To that endwe define the notion of energy of U . Definition 5.1.
The (cid:96) ∞ -energy λ ( U, x ) of U at x is defined by λ ( U, x ) = max u ∈ U | ux − x | . The (cid:96) ∞ -energy of U is given by λ ( U ) = inf x ∈ X λ ( U, x ) . A point q ∈ X is (almost-)minimising the (cid:96) ∞ -energy if λ ( U, q ) (cid:54) λ ( U ) + δ .Let x ∈ X and A, B ⊆ X . Define U x ( A, B ) be the set of elements u ∈ U satisfying the followingconditions • | x − ux | (cid:62) · δ , • there exists a ∈ A ∩ S ( x, δ ) , such that ( x, ux ) a (cid:54) δ , • there exists b ∈ B ∩ S ( x, δ ) , such that ( u − x, x ) b (cid:54) δ .We write U x ( A ) = U x ( A, A ) , and, if there is no ambiguity, U ( A, B ) = U x ( A, B ) for short. Definition 5.2 (Quasi-centre) . A point x ∈ X is a quasi-centre for U if, for all y ∈ S ( x, δ ) , we have (cid:12)(cid:12) U x (cid:0) y +100 δ (cid:1)(cid:12)(cid:12) (cid:54) | U | . Proposition 5.3.
Let q be a point that minimises the (cid:96) ∞ -energy of U . There exists a quasi-centre p for U such that | p − q | (cid:54) λ ( U ) . Remark 5.4.
The existence of a quasi-centre is already known by [DS20]. The authors prove there thatany point minimising the (cid:96) -energy is a quasi-centre. However such a point could be very far from anypoint minimising the (cid:96) ∞ -energy. Proof.
We describe a recursive procedure to find a quasi-centre p . The idea is to construct a quasi-geodesicfrom q to a quasi-centre p . Let x = q and suppose that x , . . . , x i − , x i ∈ X are already defined. If x i isa quasi-centre for U , we let p = x i and stop the induction. Otherwise, there is a point x i +1 ∈ S ( x i , δ ) such that | U x i ( x +100 δi +1 ) | > | U | . Our idea is to apply Lemma 2.2 to the sequence of points x , x , . . . , x i , x i +1 , ux i +1 , ux i , . . . , ux ,ux for some u ∈ U x i ( x +100 δi +1 ) . Like this we can write the distance from x to ux as a function of theindex i . We will observe that this function diverges to infinity, which forces the procedure to stop. To dothis, we collect the following observations. By construction, we have: Lemma 5.5.
For all u ∈ U x i − ( x +100 δi ) , the following holds(1) ( x i − , ux i − ) x i (cid:54) δ and ( x i − , ux i − ) ux i (cid:54) δ (2) ( x i − , ux i ) x i (cid:54) δ and ( x i , u x i − ) ux i (cid:54) δ . Roughly speaking, this lemma tells us that x i − , x i , ux i and ux i − are aligned in this way along theneigbourhood of the geodesic [ x i − , ux i − ] . Proof.
The first point is just a reformulation of the definition of the set U x i − ( x +100 δi ) . By construction | ux i − − x i − | > | x i − x i − | + 2 δ , for every u in this set. The second now follows from the four pointinequality (2). (cid:3) Lemma 5.6. If x i is not a quasi-centre for U , then ( x i − , x i +1 ) x i (cid:54) δ .Proof. We note that | U x i − ( x +100 δi ) ∩ U x i ( x +100 δi +1 ) | > | U | / . Let us fix an element u in this intersection.By Lemma 5.5, ( x i − , ux i ) x i (cid:54) δ and ( x i , ux i ) x i +1 (cid:54) δ . According to the four point inequalitywe have δ (cid:62) ( x i − , ux i ) x i (cid:62) min { ( x i − , x i +1 ) x i , ( x i +1 , ux i ) x i } − δ. Since | x i − x i +1 | > δ , the minimum cannot be achieved by ( x i +1 , ux i ) x i , whence the result. (cid:3) Lemma 5.7. If x i is not a quasi-centre, then, for all u ∈ U x i ( x +100 δi +1 ) , | ux − x | (cid:62) | x i +1 − ux i +1 | + 10 ( i + 1) δ. Proof.
Let u be in U x i ( x +100 δi +1 ) . By Lemma 5.5, ( x i , ux i +1 ) x i +1 (cid:54) δ and ( x i +1 , ux i ) ux i +1 (cid:54) δ .On the other hand, by Lemma 5.6, we have ( x j − , x j +1 ) x j (cid:54) δ and ( ux j − , ux j +1 ) ux j (cid:54) δ , forall < j (cid:54) i . The claim follows from Lemma 2.2 applied to the sequence of points x , x , . . . , x i , x i +1 ,ux i +1 , ux i , . . . , ux , ux . (cid:3) Suppose that x i is not a quasi-centre. Fix u ∈ U x i ( x +100 δi +1 ) . By construction we have | x i +1 − ux i +1 | (cid:62) · δ . Recall that x = q (almost) minimises the energy. By Lemma 5.7, we get λ ( U ) (cid:62) ( i + 5) δ. This means that the induction used to build the sequence ( x i ) stop after finitely many step. Moreover, whenthe process stop we have x i = p and λ ( U ) (cid:62) ( i +5) δ . For every j (cid:54) i − we have | x j − x j +1 | (cid:54) δ ,thus | p − q | (cid:54) λ ( U ) . (cid:3) RODUCT SET GROWTH IN BURNSIDE GROUPS 19
6. S
ETS OF DIFFUSE ENERGY
In this section we assume that the action of G on X is ( N, κ ) -acylindrical, with κ > · δ . Let U ⊂ G be a finite subset. Let p be a quasi-centre of U . In this section we assume that U is of diffuseenergy (at p ) that is for at least of the elements of U ⊂ G , we have | up − p | > κ .6.1. Reduction lemma.
We first prove the following variant of the reduction lemmas in [DS20].
Proposition 6.1 (Reduction) . There is v ∈ U , and U ⊂ U of cardinality | U | (cid:62) | U | such that for all u ∈ U , • ( u − p, vp ) p (cid:54) δ and ( v − p, u p ) p (cid:54) δ , and • κ (cid:54) | u p − p | (cid:54) | vp − p | . Remark 6.2.
In the case of trees, Proposition 6.1 follows directly from [DS20, Lemma 6.4], and theproof of this lemma is due to Button [But13]. The situation is different in the case of hyperbolic spaces.Indeed, in contrast to the reduction lemmas in [DS20, Section 6.1], the cardinality of U in Proposition 6.1does not depend the cardinality of balls in X , as in [DS20, Lemma 6.3], and the estimates on the Gromovproducts do not depend on the logarithm of the cardinality of U , as in [DS20, Lemma 6.8]. Proof.
For simplicity we let η = 1 / . Let U (cid:48) = { u ∈ U | | up − p | (cid:62) κ } . As the energy of U isdiffuse at p , we have | U (cid:48) | (cid:62) (1 − η ) | U | . Let us fix u ∈ U (cid:48) such that | u p − p | is maximal. We claim thefollowing result. Lemma 6.3.
At least one of the following holds:(1) there is U ⊂ U (cid:48) of cardinality | U | > η | U | such that for all u ∈ U , ( u − p, u p ) p (cid:54) δ and ( u − p, u p ) p (cid:54) δ ; (2) there are U , U ⊂ U (cid:48) of cardinalities | U | > η | U | and | U | > η | U | such that for all u ∈ U , u ∈ U , ( u − p, u p ) p (cid:54) δ and ( u − p, u p ) p (cid:54) δ. We postpone for the moment the proof of this lemma and complete first the demonstration of Propo-sition 6.1. In case (1) of Lemma 6.3, we set v = u . In case (2) of Lemma 6.3, we may assume, up toexchanging the roles of U and U , that there is v ∈ U such that for all u ∈ U , | u p − p | (cid:54) | vp − p | .This yields Proposition 6.1. (cid:3) Proof of Lemma 6.3.
We write S = S ( p, δ ) for short. For simplicity, in this proof we write U (cid:48) ( A, B ) = U (cid:48) ∩ U p ( A, B ) . See Section 5 for the definition or U p ( A, B ) .The definition of hyperbolicity implies the following useful lemma. Lemma 6.4 (Lemma 6.1 of [DS20]) . Let y , z , y , z ∈ S . If | z − y | > δ , then for every u ∈ U ( y , z ) and u ∈ U ( y , z ) , we have ( u − p, u p ) p (cid:54) δ . (cid:3) By construction | u p − p | (cid:62) · δ . Thus there exists y and z ∈ S such that ( u p, p ) y (cid:54) δ and ( p, u − p ) z (cid:54) δ , so that u ∈ U (cid:48) ( y , z ) . Assume first that | U (cid:48) ( S \ z +6 δ , S \ y +6 δ ) | > η | U | . Then welet U = U (cid:48) ( S \ z +6 δ , S \ y +6 δ ) . Using Lemma 6.4 we conclude that (1) holds.Observe that the complement in U (cid:48) of the previous set is the union of U (cid:48) ( z +6 δ , S ) and U (cid:48) ( S, y +6 δ ) .Recall that | U (cid:48) | > (1 − η ) | U | . Thus we can now assume that(6) (cid:12)(cid:12) U (cid:48) ( z +6 δ , S ) ∪ U (cid:48) ( S, y +6 δ ) (cid:12)(cid:12) > (1 − η ) | U | . Let us now assume that | U (cid:48) ( z +6 δ , S \ z +12 δ ) > η | U | . In this case we let U = U = U (cid:48) ( z +6 δ , S \ z +12 δ ) .Using Lemma 6.4 we conclude that (2) holds. The same argument works if | U (cid:48) ( S \ y +12 δ , y +6 δ ) | > η | U | .Suppose now that the cardinality of U (cid:48) ( z +6 δ , S \ z +12 δ ) and U (cid:48) ( S \ y +12 δ , y +6 δ ) are both bounded aboveby η | U | . It follows from (6) that(7) | U ∪ U | > (1 − η ) | U | , where U = U (cid:48) ( z +6 δ , z +12 δ ) and U = U (cid:48) ( y +12 δ , y +6 δ ) . Since p is a quasi-centre, the cardinality of both U and U is bounded above by | U | / . It followsfrom (7) that each of them contains at least (1 / − η ) | U | elements. Observe also that | y − z | > δ .Indeed otherwise both U and U are contained in U ( y +100 δ ) . Hence (7) contradicts the fact that p is aquasi-centre. Applying Lemma 6.4 we conclude that U and U satisfy (2). (cid:3) Construction of free sub-semi-groups.
We recall that λ ( U ) denotes the (cid:96) ∞ -energy of the finitesubset U ⊂ G . By Proposition 5.3, we can assume that the quasi-centre p , which we fixed at the beginningof this section, is at distance at most λ ( U ) from a point (almost-)minimising the (cid:96) ∞ -energy of U . We stillassume that the energy of U is diffuse (at p ). We treat p as the base point of X . Remark 6.5.
According to the triangle inequality, we have | up − p | (cid:54) λ ( U ) + δ , for every u ∈ U .Since the energy of U is diffuse at p , there is an element u ∈ U that moves p by a large distance. As aconsequence λ ( U ) (cid:62) δ , and thus | up − p | (cid:54) λ ( U ) , for every u ∈ U . This estimates are far from beingoptimal, but sharp enough for our purpose. Proposition 6.6.
There exists a δ -strongly reduced subset S ⊂ U such that | S | (cid:62) N δλ ( U ) | U | . Proof.
For simplicity we let α = 1002 δ . We fix U and v given by Proposition 6.1. We set T = U v .Using hyperbolicity we observe the followings • ( t − p, t (cid:48) p ) (cid:54) α , for all t, t (cid:48) ∈ T . • | tp − p | > α + 200 δ , for all t ∈ T . RODUCT SET GROWTH IN BURNSIDE GROUPS 21
For every s ∈ T , we set A s = { t ∈ T | | p − tp | (cid:54) | p − sp | and ( sp, tp ) p (cid:62) | tp − p | − α − δ } In order to define S , we construct by induction an increasing sequence ( S i ) of subsets of T . We first let S = ∅ . Assume that now that S i has beed defined for some integer i (cid:62) . If the set T \ (cid:91) s ∈ S i A s is empty, then the process stops and we let S = S i (note that this will ineluctably happen as T isfinite). Otherwise, we choose an element s i +1 in this set for which | p − s i +1 p | is maximal and let S i +1 = S i ∪ { s i +1 } . Lemma 6.7.
The set S is α -strongly reduced.Proof. By construction T (hence S ) is α -reduced. It suffices to prove that for every distinct s, s (cid:48) ∈ S wehave ( sp, s (cid:48) p ) p (cid:54) min {| sp − p | , | s (cid:48) p − p |} − α − δ. Using the notation above, we write, s , s , . . . , s n for the elements S in the order they have beenconstructed. Let i, j ∈ { , . . . , n } such that | p − s j p | (cid:54) | p − s i p | . If i < j , then s j does not belong to A s i , thus ( s i p, s j p ) p < | s j p − p | − α − δ (cid:54) min {| s i p − p | , | s j p − p |} − α − δ. Assume now that j < i . Note that the sequence {| p − s k p |} is non-increasing, hence | p − s j p | = | p − s i p | .Since s i does not belong to A s j , thus ( s i p, s j p ) p < | s i p − p | − α − δ (cid:54) min {| s i p − p | , | s j p − p |} − α − δ. (cid:3) Lemma 6.8.
There is a constant c > depending only on N and δ such that for every s ∈ T , we have | A s | (cid:54) λ ( U ) /c .Proof. Let s ∈ T . Let m (cid:62) be an integer. Let t = u v , t = u v , . . . , t m = u m v be m pairwisedistinct elements in A s . We focus first on a particular case and assume in addition that | u i p − u j p | > · δ for every i, j ∈ { , . . . , m } . Up to reindexing the elements we can suppose that | u p − p | (cid:54) | u p − p | (cid:54) . . . (cid:54) | u m p − p | . Since t i belongs to A s , we have ( p, sp ) t i p (cid:54) | t i p − p | − ( t i p, sp ) p (cid:54) α + 150 δ (cid:54) δ. On the other hand, we know by construction of U and v that ( u − i p, vp ) p (cid:54) δ . Hence by triangleinequality ( p, sp ) u i p (cid:54) δ . Using hyperbolicity we get ( p, u p ) u p (cid:54) δ, and ( u i − p, u i +1 p ) u i p (cid:54) δ ∀ i ∈ { , . . . , m − } . Since | u i +1 p − u i p | > · δ for every i ∈ { , . . . , m − } , it follows from Lemma 2.2 that λ ( U ) (cid:62) | u m p − p | (cid:62) m − (cid:88) i =0 | u i +1 p − u i p | − mδ + 2 δ (cid:62) mδ In particular m (cid:54) λ ( U ) /δ .Let us now estimate the cardinality of A s . Any element t ∈ A s can be written t = u t v with u t ∈ U .Consider now t, t (cid:48) ∈ A s .We claim that | u t vp − u t (cid:48) vp | (cid:54) | u t p − u t (cid:48) p | +4308 δ . Indeed, by definition of A s , we have ( p, sp ) u t vp (cid:54) α + 150 δ and ( p, sp ) u t (cid:48) vp (cid:54) α + 150 δ . It follows from hyperbolicity that | u t vp − u t (cid:48) vp | (cid:54) || p − u t vp | − | p − u t (cid:48) vp || + 2308 δ. See for instance [Cou14, Lemma 2.2(2)]. We recall that ( u − t p, vp ) p (cid:54) δ . Thus | p − u t vp | (cid:62) | p − u t p | + | vp − p | − · δ . Similarly, | p − u t (cid:48) vp | (cid:62) | p − u t (cid:48) p | + | vp − p | − · δ . This and thetriangle inequality implies | u t vp − u t (cid:48) vp | (cid:54) || p − u t p | − | p − u t (cid:48) p || + 4308 δ. Finally, by the triangle inequality || p − u t p | − | p − u t (cid:48) p || (cid:54) | u t p − u t (cid:48) p | . This implies the claim. Recallthat | vp − p | (cid:62) κ + 24 100 δ. We let M = 1203 N . Using acylindricity – see (3) – we observe that for every t ∈ A s , there are at most M elements t (cid:48) ∈ A s such that | u t p − u t (cid:48) p | (cid:54) · δ . So we can extract a subset B ⊂ A s containing m (cid:62) | A s | /M elements such that for every t, t (cid:48) ∈ B we have | u t p − u t (cid:48) p | > · δ . It follows from theprevious discussion that m (cid:54) λ ( U ) /δ . | A s | (cid:54) Nδ λ ( U ) . (cid:3) Lemma 6.9.
The cardinality of S is bounded from below as follows: | S | (cid:62) · N δλ ( U ) | T | . Proof.
By construction, the collection of sets { A s } s ∈ S covers T . We have seen in Lemma 6.8 that thecardinality of each of them is at most · N λ ( U ) /δ . Hence the result. (cid:3) The previous lemma completes the proof of Proposition 6.6. (cid:3)
RODUCT SET GROWTH IN BURNSIDE GROUPS 23
7. S
ETS OF CONCENTRATED ENERGY
We still assume here that the action of G on X is ( N, κ ) -acylindrical, with κ > · δ . Let U ⊂ G be a finite subset and p ∈ X a base point. In this section we also assume that U has concentrated energy (at p ) that is, there exists U ⊂ U with | U | (cid:62) | U | / such that | up − p | (cid:54) κ , for all u ∈ U . The goalof the section is to prove the following statement. Proposition 7.1.
Let M = 2 κN/δ . If λ ( U, p ) > κ , then either | U | (cid:54) M or there exists an κ -strongly reduced subset S ⊂ U such that | S | (cid:62) | U | /M − .Proof. The proof follows the exact same ideas as Lemmas 5.2 and 5.3 of [DS20]. Since the energy λ ( U, p ) at p is larger than κ , there exists v ∈ U satisfying | vp − p | > κ . For every u ∈ U , we let B u = { u (cid:48) ∈ U | ( uvp, u (cid:48) vp ) p (cid:62) κ − δ } . Let us fix first an element u ∈ U . We claim that the cardinality of B u is at most M . Let m be a point in X such that | p − m | = 21 κ − δ and ( p, vp ) m (cid:54) δ . Let u (cid:48) ∈ B u . The element u (cid:48) u − moves the point up by at most κ . Observe also that u (cid:48) u − moves um by at most κ + 8 δ . Indeed the distance | up − um | islarge compare to | p − up | . It follows from the four point inequality that ( p, uvp ) um (cid:54) δ . On the otherhand || p − um | − | p − m || = || p − um | − | up − um || (cid:54) | p − up | (cid:54) κ. Similarly ( p, u (cid:48) vp ) u (cid:48) m (cid:54) δ and || p − u (cid:48) m | − | p − m || (cid:54) κ . In particular both | p − um | and | p − u (cid:48) m | are smaller that ( uvp, u (cid:48) vp ) p . It follows from hyperbolicity, see for instance [Cou14, Lemma 2.2], that | um − u (cid:48) m | (cid:54) || p − um | − | p − u (cid:48) m || + 8 δ (cid:54) κ + 8 δ, which corresponds to our announcement. Recall that | p − um | (cid:62) | p − m | − κ (cid:62) κ − δ. Recall that M = 2 κN/δ . Using acylindricity – see (3) – we get that B u contains at most M elements,which completes the proof of our claim.We now fix a maximal subset U ⊂ U such that any two distinct u , u ∈ U never belong to the samesubset B u (for any u ∈ U ). The cardinality of U is at least | U | (cid:62) | U | /M . Indeed by maximality of U , the U is covered by the collection ( B u ) u ∈ U .We claim that there is at most one element u ∈ U such that ( v − p, uvp ) p > κ . Assume on thecontrary that it is not the case. Using hyperbolicity we can find two distinct element u, u (cid:48) ∈ U such that ( uvp, u (cid:48) vp ) p (cid:62) min (cid:8) ( v − p, uvp ) p , ( v − p, u (cid:48) vp ) p (cid:9) − δ > κ − δ. Thus u (cid:48) belongs to B u which contradicts the definition of U . If | U | > M then U contains at least elements. We define then U from U by removing if necessary the element u ∈ U such that ( v − p, uvp ) p > κ . Note that | U | (cid:62) | U | /M − .We now let S = U v . We are going to prove that S is κ -strongly reduced. Note first that | sp − p | (cid:62) | vp − p | − κ > κ + 200 δ for every s ∈ S . Let s = uv and s (cid:48) = u (cid:48) v be two elements in S . It followsfrom the triangle inequality that ( s − p, s (cid:48) p ) p (cid:54) ( v − p, s (cid:48) p ) p + | up − p | (cid:54) ( v − p, s (cid:48) p ) p + 2 κ. By construction of U , no element s (cid:48) ∈ S has a large Gromov product with v − . Hence ( s − p, s (cid:48) p ) p (cid:54) κ . Thus the set S is κ -reduced. By choice of U we also have ( sp, s (cid:48) p ) p (cid:54) κ − δ for every distinct s, s (cid:48) ∈ S . Recall that min {| sp − p | , | s (cid:48) p − p |} (cid:62) | vp − p | − κ > κ. Consequently S is κ -strongly reduced. (cid:3)
8. G
ROWTH IN GROUPS ACTING ON HYPERBOLIC SPACES
As a warmup for the study of Burnside groups we first prove the following statement.
Theorem 8.1.
Let δ > , κ (cid:62) · δ , and N > . Assume that the group G acts ( N, κ ) -acylindricallyon a δ -hyperbolic length space. For every finite U ⊂ G such that λ ( U ) > κ , one of the followingholds.(1) | U | (cid:54) κN/δ .(2) There is an α -strongly reduced subset S ⊂ U with α (cid:54) κ , and cardinality | S | (cid:62) N δλ ( U ) | U | . Proof of Theorem 8.1.
Let U ⊂ G be a finite subset such that λ ( U ) > κ . Choice of the base-point.
Let q be a point minimising the (cid:96) ∞ -energy of U . We now fix the base-point p to be a quasi-centre for U . By Proposition 5.3, we can assume that | p − q | (cid:54) λ ( U ) . Case 1: diffuse energy.
Let us first assume that U is of diffuse energy at p . That is, there is a subset U (cid:48) ⊂ U such that | U (cid:48) | (cid:62) | U | / and such that for all u (cid:48) ∈ U (cid:48) we have | u (cid:48) p − p | > κ . Then, byProposition 6.6, there is a strongly reduced subset S ⊂ U of cardinality | S | (cid:62) N δλ ( U ) | U | . Case 2: concentrated energy.
Otherwise U is of concentrated energy at p . Indeed, there is a subset U (cid:48) ⊂ U of cardinality | U (cid:48) | (cid:62) | U | / such that | u (cid:48) p − p | (cid:54) κ , for all u (cid:48) ∈ U (cid:48) . Recall that λ ( U ) > κ .Assume that | U | > κN/δ . By Proposition 7.1, there is a strongly reduced subset S ⊂ U of cardinality | S | (cid:62) N δκ | U | − (cid:62) N δλ ( U ) | U | . RODUCT SET GROWTH IN BURNSIDE GROUPS 25
This completes the proof of Theorem 8.1. (cid:3)
Corollary 8.2.
Let δ > , κ (cid:62) · δ , and N > . Assume that the group G acts ( N, κ ) -acylindricallyon a δ -hyperbolic length space. For every finite U ⊂ G such that λ ( U ) > κ and for all integer r (cid:62) ,we have | U r | (cid:62) (cid:18) N δλ ( U ) | U | (cid:19) [( r +1) / . Proof.
Without loss of generality we can assume that | U | > κN/δ . Indeed otherwise the base of theexponential function on the right hand side of the stated inequality is less than one, hence the statement isvoid. According to Theorem 8.1, there exists a subset S ⊂ U which is strongly reduced with cardinality | S | (cid:62) N δλ ( U ) | U | . By Lemma 3.2, S freely generates a free-sub-semigroup of G . Hence for all integer r (cid:62) , | U r | (cid:62) | S | [( r +1) / (cid:62) (cid:18) N δλ ( U ) | U | (cid:19) [( r +1) / . (cid:3) We now combine Theorem 8.1 with our estimates on the growth of aperiodic words, see Proposition3.11. If we use Proposition 4.2 to compare the notion of aperiodic words and power-free elements weobtain the following useful growth estimate.
Corollary 8.3.
Let δ > , κ (cid:62) · δ , N > and λ (cid:62) . There exists a parameter m > withthe following properties. Assume that the group G acts ( N, κ ) -acylindrically on a δ -hyperbolic geodesicspace. Let U ⊂ G such that κ < λ ( U ) (cid:54) λ . One of the following holds.(1) | U | (cid:54) max { κN/δ, · N λ ( U ) /δ } .(2) Given r > and m (cid:62) m , denote by K ( m, r ) the set of all m -power-free elements in U r . Then, | K ( m, r ) | (cid:62) (cid:18) · N δλ ( U ) | U | (cid:19) r . Proof.
Let U ⊂ G be a finite subset such that λ ( U ) > κ . Without loss of generality we can assumethat | U | > max { κN/δ, · N λ ( U ) /δ } . By Theorem 8.1 there is a α -strongly reduced subset S ⊂ U where α (cid:54) κ and such that | S | (cid:62) N δλ ( U ) | U | . It follows from our choice that | S | (cid:62) and λ ( S ) (cid:54) λ ( U ) . In view of Proposition 3.11, there exists m > , which only depends on δ , N , κ and λ ( U ) such that for every m (cid:62) m , for every r (cid:62) , we have,(8) | S ∗ m ∩ W ( r + 1) | (cid:62) | S | | S ∗ m ∩ W ( r ) | . (Recall that W ( r ) is the ball of radius r , for the word metric with respect of S and S ∗ m is the set of m -aperiodic words over S .) Let us now focus on the cardinality of spheres in S ∗ m . As S is α -strongly reduced, it generates a free sub-semi-group (Lemma 3.2). Thus | S r +1 m | = | S ∗ m ∩ W ( r + 1) | − | S ∗ m ∩ W ( r ) | . If we combine this inequality (8) and the fact that | S | / (cid:62) , we obtain that | S r +1 m | (cid:62) | S | | S ∗ m ∩ W ( r ) | − | S ∗ m ∩ W ( r ) | (cid:62) (cid:18) | S | − (cid:19) | S ∗ m ∩ W ( r ) | (cid:62) | S | | S ∗ m ∩ W ( r ) | (cid:62) | S | | S rm | . By an inductive argument, we obtain that, for all r (cid:62) , | S rm | (cid:62) (cid:18) | S | (cid:19) r . Now let m = m + (2 λ ( S ) + 20 δ ) /τ and let m (cid:62) m . Then, by Proposition 4.2, the set S rm (cid:48) iscontained in K ( m, r ) , where m (cid:48) = m − (2 λ ( S ) + 20 δ ) /τ is larger than m . Thus, | K ( m, r ) | (cid:62) | S rm (cid:48) | (cid:62) (cid:18) | S | (cid:19) r . This completes the proof. (cid:3)
9. S
MALL CANCELLATION GROUPS
In this section we recall the necessary background on small cancellation theory with a special attentionon acylindricity, see Proposition 9.9. The presentation follows [Cou14] in content and notations.9.1.
Cones.
Let Y be a metric length space and let ρ > . The cone of radius ρ over Y is the set Z ( Y ) = Y × [0 , ρ ] / ∼ , where ∼ is the equivalence relation which identifies all the points of the form ( y, for y ∈ Y . If x ∈ Z ( Y ) , we write x = ( y, r ) to say that ( y, r ) represents x . We let v = ( y, be the apex of the cone.If y , y (cid:48) are in Y , we let θ ( y, y (cid:48) ) = min { π, | y − y (cid:48) | / sinh ρ } be their angle at v . There is a metric on Z ( Y ) that is characterised as follows, see [BH99, Chapter I.5]. Let x = ( y, r ) and x (cid:48) = ( y (cid:48) , r (cid:48) ) in Z ( Y ) .Then cosh | x − x (cid:48) | = cosh r cosh r (cid:48) − sinh r sinh r (cid:48) cos θ ( y, y (cid:48) ) . It turns out that Z ( Y ) is a hyperbolic space [Cou14, Proposition 4.6].We let ι : Y → Z ( Y ) be the embedding defined as ι ( y ) = ( y, ρ ) . The metric distortion of ι iscontrolled by a function µ : R + → [0 , ρ ] that is characterised as follows: for every t ∈ R + , cosh µ ( t ) = cosh ρ − sinh ρ cos (cid:18) min (cid:26) π, t sinh ρ (cid:27)(cid:19) . For all y, y (cid:48) ∈ Y , we have(9) | ι ( y ) − ι ( y (cid:48) ) | Z ( Y ) = µ ( | y − y (cid:48) | Y ) . Let us mention some properties of µ for later use. RODUCT SET GROWTH IN BURNSIDE GROUPS 27
Proposition 9.1 (Proposition 4.4 of [Cou14]) . The map µ is continuous, concave, non-decreasing. More-over, if µ ( t ) < ρ , then t (cid:54) π sinh( µ ( t ) / . (cid:3) Let H be a group that acts by isometries on Y . Then H acts by isometries on Z ( Y ) by hx = ( hy, r ) .We note that H fixes the apex of the cone.9.2. The cone off space.
From now, we assume that X is a proper, geodesic, δ -hyperbolic space, where δ > . We fix a parameter ρ > , whose value will be made precise later. In addition, we consider a group G that acts properly co-compactly by isometries on X . We assume that this action is ( N, κ ) -acylindrical.We let Q be a collection of pairs ( H, Y ) such that Y is closed strongly-quasi-convex in X and H is a subgroup of Stab( Y ) acting co-compactly on Y . Suppose that the group G acts on Q by the rule g ( H, Y ) = ( gHg − , gY ) and that Q /G is finite. Furthermore, we let ∆( Q ) = sup (cid:8) diam (cid:0) Y +5 δ ∩ Y +5 δ (cid:1) | ( H , Y ) (cid:54) = ( H , Y ) ∈ Q (cid:9) and T ( Q ) = inf {(cid:107) h (cid:107) | h ∈ H \ { } , ( H, Y ) ∈ Q } . Let ( H, Y ) ∈ Q . We denote by | · | Y the length metric on Y induced by the restriction of | · | to Y . As Y is strongly quasi-convex, for all y, y (cid:48) ∈ Y , | y − y (cid:48) | X (cid:54) | y − y (cid:48) | Y (cid:54) | y − y (cid:48) | X + 8 δ. We write Z ( Y ) for the cone of radius ρ over the metric space ( Y, | · | Y ) .We let the cone-off space ˙ X = ˙ X ( Y, ρ ) be the space obtained by gluing, for each pair ( H, Y ) ∈ Q ,the cone Z ( Y ) on Y along the natural embedding ι : Y → Z ( Y ) . We let V denote the set of apices of ˙ X . We endow ˙ X with the largest metric | · | ˙ X such that the map X → ˙ X and the maps Z ( Y ) → ˙ X are -Lipschitz, see [Cou14, Section 5.1]. It has the following properties. Lemma 9.2 (Lemma 5.7 of [Cou14]) . Let ( H, Y ) ∈ Q . Let x ∈ Z ( Y ) and x (cid:48) ∈ ˙ X . Let d ( x, Y ) be the distance from x to ι ( Y ) computed in Z ( Y ) . If | x − x (cid:48) | ˙ X < d ( x, Y ) , then x (cid:48) ∈ Z ( Y ) and | x − x (cid:48) | ˙ X = | x − x (cid:48) | Z ( Y ) . We recall that µ is the map that controls the distortion of the embedding ι of Y in its cone, see (9). Italso controls the distortion of the map X → ˙ X . Lemma 9.3 (Lemma 5.8 of [Cou14]) . For all x, x (cid:48) ∈ X , we have µ ( | x − x (cid:48) | X ) (cid:54) | x − x (cid:48) | ˙ X (cid:54) | x − x (cid:48) | X . (cid:3) The action of G on X then extends to an action by isometries on ˙ X : given any g ∈ G , a point x = ( y, r ) in Z ( Y ) is sent to the point gx = ( gy, r ) in Z ( gY ) . We denote by K the normal subgroup generated bythe subgroups H such that ( H, Y ) ∈ Q . The quotient space.
We let X = ˙ X/K and G = G/K . We denote by ζ the projection of ˙ X onto X and write x for ζ ( x ) for short. Furthermore, we denote by V the image in X of the apices V . Weconsider X as a metric space equipped with the quotient metric, that is for every x, x (cid:48) ∈ ˙ X | x − x (cid:48) | X = inf h ∈ K | hx − x (cid:48) | ˙ X . We note that the action of G on ˙ X induces an action by isometries of G on X . The following theoremsummarises Proposition 3.15 and Theorem 6.11 of [Cou14]. Theorem 9.4 (Small Cancellation Theorem [Cou14]) . There are distances δ , δ , ∆ and ρ (that do notdepend on X or Q ) such that, if δ (cid:54) δ , ρ > ρ , ∆( Q ) (cid:54) ∆ , and T ( Q ) > π sinh ρ , then the followingholds:(1) X is a proper geodesic δ -hyperbolic space on which ¯ G acts properly co-compactly.(2) Let r ∈ (0 , ρ/ . If for all v ∈ V , the distance | x − v | (cid:62) r then the projection ζ : ˙ X → X induces an isometry from B ( x, r ) onto B ( x, r ) .(3) Let ( H, Y ) ∈ Q . If v ∈ V stands for the apex of the cone Z ( Y ) , then the projection from G onto G induces an isomorphism from Stab( Y ) /H onto Stab( v ) . (cid:3) Let us now fix δ , δ , ∆ and ρ as in Theorem 9.4. We assume that δ (cid:54) δ , ∆( Q ) (cid:54) ∆ , T ( Q ) > π sinh ρ , and ρ > ρ , so that X is δ -hyperbolic, with δ (cid:54) δ .We use point (2) of Theorem 9.4 to compare the local geometry of ˙ X and X . To compare the globalgeometry, we use the following proposition. Proposition 9.5 (Proposition 3.21 of [Cou14]) . Let Z ⊂ X be δ -quasi-convex and d (cid:62) δ . If, forall v ∈ V , we have Z ∩ B ( v, ρ/ d + 1210 δ ) = ∅ , then there is a pre-image Z ⊂ ˙ X such that theprojection ζ induces an isometry from Z onto Z .In addition, if S ⊂ G such that S Z ⊆ Z + d , then there is a pre-image S ⊂ G such that for every g ∈ S , z, z (cid:48) ∈ Z , we have | g z − z (cid:48) | = | gz − z (cid:48) | ˙ X . (cid:3) Group action on X . We collect some properties of the action of G . Lemma 9.6 (Lemma 6.8 of [Cou14]) . If v ∈ V and g ∈ G \ Stab( v ) , then for every x ∈ X we have | g x − x | (cid:62) ρ − | x − v | ) . (cid:3) In combination with assertion (2) of Theorem 9.4, the previous lemma implies that local properties ofthe action are often inherited from the action of G on the cone-off space. For example, if F is an ellipticsubgroup of G , then either F ⊆ Stab( v ) for some v ∈ V or it is the image of an elliptic subgroup of G ,see [Cou14, Proposition 6.12]. RODUCT SET GROWTH IN BURNSIDE GROUPS 29
There is a lower bound on the injectivity radius of the action on X , and an upper bound on theacylindricity parameter. Proposition 9.7 (Proposition 6.13 of [Cou14]) . Let (cid:96) = inf {(cid:107) g (cid:107) ∞ | g (cid:54)∈ Stab( Y ) , ( H, Y ) ∈ Q} . Then τ ( G, X ) (cid:62) min (cid:26) ρ(cid:96) π sinh ρ , δ (cid:27) . (cid:3) We recall that L is the number fixed in Section 2.2 using stability of quasi-geodesics. Proposition 9.8 (Corollary 6.15 of [Cou14]) . Assume that all elementary subgroups of G are cyclicinfinite or finite with odd order. If Stab( Y ) is elementary for every ( H, Y ) ∈ Q , then A ( G, X ) (cid:54) A ( G, X ) + 5 π sinh(2 L δ ) . (cid:3) Note that the proposition actually does not require that finite subgroups of G have odd order. Thisassumption in [Cou14, Propositions 6.15] was mainly made to simplify the overall exposition in this paper.The mistake of the order of π sinh(2 L δ ) in the above estimates is reminiscent of the distortion of theembedding of X into ˙ X , measured by the map µ , see Proposition 9.1.9.5. Acylindricity.
Let us assume that all elementary subgroups of G are cyclic. In particular, it followsthat ν ( G, X ) = 1 , see for instance [Cou14, Lemma 2.40]. Moreover, we assume that for every pair ( H, Y ) ∈ Q , there is a primitive hyperbolic element h ∈ G and a number n such that H = (cid:104) h n (cid:105) and Y isthe cylinder C H of H . Proposition 9.9.
The action of G on X is ( N , κ ) -acylindrical, where N (cid:54) max (cid:26) N, π sinh ρτ ( G, X ) + 1 (cid:27) and κ = max { A ( G, X ) , κ } + 5 π sinh(150¯ δ ) . Remark 9.10.
It is already known that if G acts acylindrically on X , then so does G on X , see Dahmani-Guirardel-Osin [DGO17, Proposition 2.17, 5.33]. However in their proof κ is much larger than ρ . For ourpurpose we need a sharper control on the acylindricity parameters. With our statement, we will be able toensure that κ (cid:28) ρ .Later we will use this statement during an induction process for which we also need to control uniformly the value of N . Unlike in [DGO17], if N is very large, our estimates tells us that ¯ N (cid:54) N . Proof.
Let S ⊂ G , let Z = Fix( S, δ ) and let us assume that diam Z (cid:62) κ . We are going to prove that S contains at most N elements. Wedistinguish two cases: either S fixes an apex v ∈ V or not. Lemma 9.11.
If there is v ∈ V , such that S ⊂ Stab( v ) , then | S | (cid:54) π sinh ρ/τ ( G, X ) + 1 . Proof. If S ⊂ Stab( v ) , then v ∈ Z . As diam( Z ) (cid:62) κ , there is a point z ∈ Z +10¯ δ such that z ∈ B ( v, ρ/ and | z − v | > δ . Moreover, for all s ∈ S , we have | s z − z | (cid:54) δ . Let v be a pre-image of v and z a pre-image of z in the ball B ( v, ρ/ . For every s ∈ S , we choose a pre-image s ∈ G such that | sz − z | ˙ X (cid:54) δ and write S for the set of all pre-images obtained in this way. Observe that by thetriangle inequality, | sv − v | ˙ X (cid:54) ρ + 120 δ , for every s ∈ S . However any two distinct apices in ˙ X areat a distance at least ρ . Thus S is contained in Stab( v ) . If ( H, Y ) ∈ Q such that v is the apex of thecone Z ( Y ) , then, by Lemma 9.2, | sz − z | Z ( Y ) (cid:54) δ < | z − v | Z ( Y ) + | sz − v | Z ( Y ) . Let y be a radialprojection of z on Y . By the very definition of the metric on Z ( Y ) , we get that | sy − y | < π sinh ρ .Recall that every elementary subgroup is cyclic, in particular so is Stab( Y ) . Consequently the number ofelements g ∈ Stab( Y ) such that | gy − y | (cid:54) r is linear in r . More precisely, using Lemma 2.12, we have | S | (cid:54) π sinh ρ + 112 δ ) τ ( G, X ) + 1 (cid:54) π sinh ρτ ( G, X ) + 1 , which yields the claim. (cid:3) Lemma 9.12. If S does not stabilise any v ∈ V , then | S | (cid:54) N .Proof. By Lemma 9.6, Z ∩ B ( v, ρ − δ ) = ∅ , for every v ∈ V . By Lemma 2.4, Z is δ -quasi-convex.By Lemma 9.5, there exists pre-images Z ⊂ ˙ X and S ⊂ G such that diam( Z ) > κ and for all s ∈ S andall z ∈ Z , we have | sz − z | ˙ X (cid:54) δ .Let us write d = π sinh(150¯ δ ) . We now focus on the subset Fix(
S, d ) ⊂ X . Let x, y ∈ Z such that | x − y | > κ . Let p , q be projections of x , y in X . Then, as | p − x | ˙ X (cid:54) δ and | q − y | ˙ X (cid:54) δ , | p − q | ˙ X (cid:62) κ − δ . As | p − q | X (cid:62) | p − q | ˙ X , the distance | p − q | X > κ − δ. On the otherhand, µ ( | sp − p | X ) (cid:54) | sp − p | ˙ X < δ < ρ . Thus, by Proposition 9.1, | sp − p | X < d. Similarly, | sq − q | X < d . This means that the diameter of Fix(
S, d ) ⊂ X is larger than κ − δ , hence, larger than A ( G, X ) + 4 d + 209 δ . It follows by Proposition 2.10 that S generates an elementary subgroup E .Suppose first that this subgroup E is loxodromic. It is infinite cyclic by assumption. Recall that thetranslation length of any element in S is at most d . Hence, as previously we get | S | (cid:54) d + 112 δ ) τ ( G, X ) + 1 (cid:54) π sinh ρτ ( G, X ) + 1 . Suppose now that E is an elliptic subgroup. In particular, the set Fix( S, δ ) ⊂ X is non-empty, and byLemma 2.4, Fix(
S, d ) is contained in the d/ -neighbourhood of Fix( S, δ ) . In particular the diameter of Fix( S, δ ) is larger that κ − δ − d , hence, larger than κ . Consequently by acylindricity, | S | (cid:54) N . (cid:3) This completes the proof of Proposition 9.9. (cid:3) (cid:96) ∞ -energy. In this section we compare the (cid:96) ∞ -energy of finite subset U ⊂ G and its image U ⊂ G respectively. RODUCT SET GROWTH IN BURNSIDE GROUPS 31
Proposition 9.13.
Let U ⊂ G be a finite set such that λ ( U ) (cid:54) ρ/ . If, for all v ∈ V , the set U is notcontained in Stab( v ) , then there is a pre-image U ⊂ G of U of energy λ ( U ) (cid:54) π sinh λ ( U ) . Proof.
Let (cid:15) > . Let q ∈ X such that λ ( U , q ) (cid:54) λ ( U )+ (cid:15) . By Lemma 9.6, | q − v | > ρ − ( λ ( U )+ (cid:15) ) / > ρ/ , for all v ∈ V . Let q be a pre-image of q in ˙ X . We choose a pre-image U ⊂ G of U such that forevery u ∈ U , we have | uq − q | ˙ X = | uq − q | . Let x ∈ X be a projection of q onto X . We note that µ ( | ux − x | X ) (cid:54) | ux − x | ˙ X (cid:54) λ ( U ) + (cid:15) ) < ρ . Thus | ux − x | X (cid:54) π sinh( λ ( U ) + (cid:15) ) , see Proposition9.1. We just proved that λ ( U ) (cid:54) π sinh( λ ( U ) + (cid:15) ) for every (cid:15) > , whence the result. (cid:3)
10. P
RODUCT SET GROWTH IN B URNSIDE GROUPS OF ODD EXPONENT
We finally prove Theorem 1.1.10.1.
The induction step.
We will use the following.
Proposition 10.1 (cf. Proposition 6.18 of [Cou14]) . There are distances ρ , δ > , and A ∈ [25 · δ , ρ / , as well as natural numbers L and n such that the following holds.Let n (cid:62) n and n (cid:62) n be an odd integer. Let G act properly co-compactly by isometries on a propergeodesic δ -hyperbolic space X such that(1) the elementary subgroups of G are cyclic or finite of odd order n ,(2) A ( G, X ) (cid:54) A and τ ( G, X ) (cid:62) (cid:112) ρ L δ / n , and(3) the action of G is ( N, A ) -acylindrical.Let P be the set of primitive hyperbolic elements h of translation length (cid:107) h (cid:107) (cid:54) L δ . Let K be the normalclosure of the set { h n | h ∈ P } in G .Then there is proper geodesic δ -hyperbolic space X on which G = G/K acts properly co-compactlyby isometries. Moreover, • (1) and (2) hold for the action of G on X ; • the action of G on X is ( N , A ) -acylindrical where N = max { N, n } ; • if U is a subset of G with λ ( U ) (cid:54) ρ / that does not generated a finite subgroup, then thereexists a pre-image U ⊂ G of U such that λ ( U ) (cid:54) √ n sinh λ ( U ) . Remark 10.2.
Note that Assumptions (2) and (3) are somewhat redundant. Indeed, if the action of G on X is ( N, κ ) -acylindrical, then the parameters A ( G, X ) and τ ( G, X ) can be estimated in terms of δ , N and κ only. However, we chose to keep them both, to make it easier to apply existing results in theliterature. Proof.
This is essentially Proposition 7.1 of [Cou14]. The only additional observation is point (3). Fordetails of the proof, we refer the reader to [Cou14]. Here, we only give a rough idea of the proof and fixsome notation for later use.
We choose for δ , ∆ , δ , and ρ the constants given by the Small Cancellation Theorem, seeTheorem 9.4. We fix A = max (cid:8) π sinh(2 L δ ) , · δ (cid:9) . Without loss of generality we can assume that δ , ∆ (cid:28) δ while ρ (cid:29) L δ . In particular A (cid:54) ρ / .Following [Cou14, page 319], we define a rescaling constant as follows. Let ε n = 8 π sinh ρ √ ρ L δ √ n . We note for later use that is ρ is sufficiently large (which we assume here) we have ε n (cid:62) / √ n , for every n > . We then choose n such that for all n (cid:62) n , the following holds ε n δ (cid:54) δ , (10) ε n ( A + 118 δ ) (cid:54) min { ∆ , π sinh(2 L δ ) } , (11) ε n ρ L δ π sinh ρ (cid:54) δ , (12) ε n < . (13)These are the same conditions as in [Cou14, page 319] (in this reference, ε is denoted by λ ). We now fix n (cid:62) n and an odd integer n (cid:62) n . For simplicity we let ε = ε n . Moreover, let Q = (cid:8)(cid:0) (cid:104) h n (cid:105) , C E ( h ) (cid:1) | h ∈ P (cid:9) . As explained in [Cou14, Lemma 7.2], the small cancellation hypothesis needed to apply Theorem 9.4 aresatisfied by Q for the action of G on εX . We let G and X as in Section 9.3 (applied to G acting on εX ).Observe, for later use, that the map X → εX ζ −→ X is ε -Lipschitz. Assertions (1) and (2) follows from Lemmas 7.3 and 7.4 in [Cou14]. By Proposition 9.9the action of G on X is ( ¯ N , ¯ κ ) -acylindrical where N (cid:54) max (cid:26) N, π sinh ρτ ( G, εX ) + 1 (cid:27) and κ = max { A ( G, (cid:15)X ) , (cid:15)A } + 5 π sinh(150 δ ) . It follows from the definition of ε and our hypothesis on τ ( G, X ) that ¯ N (cid:54) max { N, n } . On the otherhand by (11) we have κ (cid:54) εA + 5 π sinh(150 δ ) (cid:54) A . Hence the action of the G on X is ( ¯ N , A ) -acylindrical as we announced.Consider now a subset U of G such that λ ( U ) (cid:54) ρ / and U does not generate a finite subgroup.Hence, applying Proposition 9.13, we see that there exists a pre-image U ⊂ G of U such that the (cid:96) ∞ -energy of U for the action of G on εX is bounded above by π sinh λ ( U ) . Thus, for the action of G on X ,we obtain that λ ( U ) (cid:54) ε − π sinh λ ( U ) < √ n sinh λ ( U ) . RODUCT SET GROWTH IN BURNSIDE GROUPS 33
This is the lifting property stated at the end of Proposition 10.1. (cid:3)
Assume now that G is a non-elementary, torsion-free hyperbolic group. Proposition 10.1 can be usedas the induction step to build from G a sequence of hyperbolic groups ( G i ) that converges to the infiniteperiodic quotient G/G n , provided n is a sufficiently large odd exponent. For our purpose, we need asufficient condition to detect whenever an element g ∈ G has a trivial image in G/G n . This is the goal ofthe next statement, see [Cou18a, Theorem 4.13]. The result is reminiscence of the key argument used byOl’shanski˘ı in [Ol’91, §10]. Theorem 10.3.
Let G be a non-elementary torsion-free group acting properly co-compactly by isometrieson a pointed hyperbolic geodesic space ( X, p ) . There are n and ξ such that for all odd integers n (cid:62) n the following holds. If g and g are two elements of G whose images in G/G n coincide, then one of themcontains a ( n/ − ξ ) -power. (cid:3) Here, we need a stronger result. Indeed we will have to apply this criterion for any group ( G i ) approximating G/G n . In particular we need to make sure that the critical exponent n appearing inTheorem 10.3 does not depend on i . For this reason, we use instead the following statement. Theorem 10.4.
There are distances ρ , δ > , and A ∈ [25 · δ , ρ / , as well as natural numbers L , n such that the following holds.Let n (cid:62) n and set ξ = n + 1 . Fix an odd integer n (cid:62) max { , n } . Let G be a group actingproperly, co-compactly by isometries on a proper, geodesic, δ -hyperbolic, pointed space ( X, p ) such that(1) the elementary subgroups of G are cyclic or finite of odd order n ,(2) A ( G, X ) (cid:54) A and τ ( G, X ) (cid:62) (cid:112) ρ L δ / n .If g and g are two elements of G whose images in G/G n coincide, then one of them contains a ( n/ − ξ ) -power. Remark 10.5.
The “novelty” of Theorem 10.4 compared to Theorem 10.3 is that the critical exponent n does not depend on G but only on the parameters of the action of G on X (acylindricity, injectivity radius,etc). Note that the critical exponent given by Ol’shanski˘ı in [Ol’91] only depends on the hyperbolicityconstant of the Cayley graph of G . However this parameter will explode along the sequence ( G i ) . Thuswe cannot formaly apply this result. Although it is certainly possible to adapt Ol’shanski˘ı’s method, werely here on the material of [Cou18a]. Sketch of proof.
The arguments follow verbatim the ones of [Cou18a, Section 4]. Observe first that theparameters δ , L , ρ , A and n in [Cou18a, p. 797] are chosen in a similar way as we did in the proof ofProposition 10.1 (note that the rescaling parameter that denote ε n is called λ n there). Once n (cid:62) n hasbeen fixed, we set, exactly as in [Cou18a, p. 797], ξ = n + 1 and n = max { , n } . We now fix an odd integer n (cid:62) n . At this point in the proof of [Cou18a] one chooses a non-elementary torsion-free group G acting properly co-compactly on a pointed hyperbolic space ( X, p ) . Note in particular that the base point p is chosen after fixing all the other parameters. Next one uses an analogue of Proposition 10.1 to build asequence of hyperbolic groups ( G i ) converging to G/G n . The final statement, that is Theorem 10.3, isthen proved using an induction on i , see [Cou18a, Proposition 4.6].Observe that the fact that G is torsion-free is not necessary here. We only need that the initial group G satisfies the induction hypothesis, that is:(1) X is a geodesic δ -hyperbolic space on which G acts properly co-compactly by isometries.(2) the elementary subgroups of G are cyclic or finite of odd order n ,(3) A ( G, X ) (cid:54) A and τ ( G, X ) (cid:62) (cid:112) ρ L δ / n .These are exactly the assumptions stated in Theorem 10.4. In particular, we can build as in [Cou18a] asequence of hyperbolic ( G i ) converging to G/G n . The theorem is proved using an induction on i just asin [Cou18a]. Actually the proof is even easier, since we only need a sufficient condition to detect elementsof G which are not trivial in G/G n , while [Cou18a] provides a sufficient and necessary condition for thisproperty. (cid:3) The approximating sequence.
Let G be a non-elementary torsion-free hyperbolic group. Theperiodic quotient G/G n is the direct limit of a sequence of infinite hyperbolic groups G i that can berecursively constructed as follows. We let δ , ρ , L , n , and A (cid:62) · δ be the parameters given byProposition 10.1.Let G = G and let X be its Cayley graph. Up to rescaling X we can assume that X is a δ -hyperbolic metric geodesic space and A ( G , X ) (cid:54) A . We choose n (cid:62) n such that τ ( G , X ) (cid:62) (cid:114) ρ L δ n . Recall that the action of G on X is proper and co-compact. Thus there exists N (cid:62) n , such thatevery subset S ⊂ G for which Fix( S, δ ) is non-empty contains at most N elements. Consequentlythe action is ( N, A ) -acylindrical. For simplicity we let λ = √ n π sinh (100 A ) and denote by m = m ( δ , N, A , λ ) the parameter given by Corollary 8.3. In addition, we set ξ = n + 1 and n = max { , n , m + ξ ) } . Let n (cid:62) n be an odd integer. It follows from our choices that the assumptions of Proposition 10.1 arethen satisfied for the action of G on X .Let us suppose that G i is already given, and acts on a δ -hyperbolic space X i such that the assumptionsof Proposition 10.1 are satisfied. Then G i +1 = G i and X i +1 = X i are given by Proposition 10.1. Notethat we have chose N in such a way that since the action of G i on X i is ( N, A ) -acylindrical, then so is RODUCT SET GROWTH IN BURNSIDE GROUPS 35 the one of G i +1 on X i +1 . It follows from the construction that G/G n is the direct limit of the sequence ( G i ) . Compare with [Cou14, Theorem 7.7]. Remark 10.6.
As the quotient
G/G n is a direct limit of non-elementary hyperbolic groups, it is an infinitegroup itself. In fact, for the same reason, it is not finitely presented either, see [Cou14, Theorem 7.7].10.3. Growth estimates.
As before, we write ε = ε n for the renormalisation parameter that we used inthe proof of Proposition 10.1. The action of G on X is proper and co-compact, hence there exists aninteger M such that for every x ∈ X , |{ g ∈ G | | gx − x | (cid:54) A }| (cid:54) M . We now let M = max (cid:26) M , A Nδ , · N λ δ (cid:27) , and a = 1 M .
Let V ⊂ G/G n be finite and not contained in a finite subgroup. Recall that if U i ⊂ G i is a pre-imageof V , its energy measured in X i is defined by λ ( U i ) = inf x ∈ X i max u ∈ U i | ux − x | X i . We now let(14) j = inf { i ∈ N | there is a pre-image U ⊂ G i of V such that λ ( U ) (cid:54) A } . Recall that the map X i → X i +1 is ε -Lipschitz. Hence, if U i +1 ⊂ G i +1 is the image a of a subset U i ⊂ G i ,we have λ ( U i +1 ) (cid:54) ελ ( U i ) . Since ε < , the index j is well-defined. Let us fix a pre-image U j of V in G j such that λ ( U j ) (cid:54) A . We now distinguish two cases. Case 1.
Assume that j = 0 . It follows from our choice of M , that | V | (cid:54) | U | (cid:54) M . Thus for every r (cid:62) we have | V r | (cid:62) (cid:62) (cid:18) M | V | (cid:19) [( r +1) / (cid:62) ( a | V | ) [( r +1) / . Case 2.
Assume that j > . Note that U j cannot generate a finite subgroup G j , otherwise so would V in G/G n . Recall that A (cid:54) ρ / . By Proposition 10.1, there exists a pre-image U j − ⊂ G j − of U j such that the energy of U j − satisfies λ ( U j − ) (cid:54) λ . By definition of j , we also have λ ( U j − ) > A .For simplicity we let m = n/ − ξ and denote by K ( m, r ) the set of elements in U rj − which are m -power-free, for every r (cid:62) . It follows from our choice of n that m (cid:62) m . Hence we can applyCorollary 8.3 so that one of the following holds. • The cardinality of U j − is at most M . In particular the same holds for V and we prove as inCase 1 that for every r (cid:62) , | V r | (cid:62) ( a | V | ) [( r +1) / . • For every r (cid:62) , we have | K ( m, r ) | (cid:62) (cid:18) · N δ λ | U j − | (cid:19) r (cid:62) ( a | V | ) r . According to our choice of n , we have n (cid:62) max { , n } . Moreover, by construction G j − satisfies the assumptions of Theorem 10.4. Hence, the map G j − → G/G n induces an embeddingfrom K ( m, r ) into V r . Consequently | V r | (cid:62) ( a | V | ) [( r +1) / . This completes the proof of Theorem 1.1.
Proof of Corollary 1.2.
Let n > and a > be the constants given by Theorem 1.1. We fix N such that a N > . Let n (cid:62) n . Let us take a subset V ⊂ G/G n that is not contained in a finite subgroup andthat contains the identity. Then, for all k (cid:62) , we have V k − ⊆ V k . As V is not contained in a finitesubgroup, this implies that | V k | > | V k − | . Thus a | V N | > . We now apply twice Theorem 1.1, firstwith the set V N , and second with V N . For every integer r (cid:62) , we have (cid:12)(cid:12) V rN (cid:12)(cid:12) (cid:62) (cid:0) a (cid:12)(cid:12) V N (cid:12)(cid:12)(cid:1) [( r +1) / (cid:62) (cid:16) a (cid:0) a (cid:12)(cid:12) V N (cid:12)(cid:12)(cid:1) (cid:17) [( r +1) / (cid:62) (cid:0) a | V N | · | V N | (cid:1) [( r +1) / Recall that a | V N | > . Hence, for every integer r (cid:62) , (cid:12)(cid:12) V rN (cid:12)(cid:12) (cid:62) | V N | [( r +1) / (cid:62) | V | [( r +1) / . Taking the logarithm and passing to the limit we get h ( V ) (cid:62) N ln( | V | ) . Since V does not lie in a cyclic subgroup, it contains at least two elements, whence the second inequalityin our statement. (cid:3) R EFERENCES[Adi79] S. I. Adian.
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