Rational growth and degree of commutativity of graph products
aa r X i v : . [ m a t h . G R ] J a n RATIONAL GROWTH AND DEGREE OF COMMUTATIVITYOF GRAPH PRODUCTS
MOTIEJUS VALIUNAS
Abstract.
Let G be an infinite group and let X be a finite generating set for G such thatthe growth series of G with respect to X is a rational function; in this case G is said to haverational growth with respect to X . In this paper a result on sizes of spheres (or balls) in theCayley graph Γ( G, X ) is obtained: namely, the size of the sphere of radius n is bounded aboveand below by positive constant multiples of n α λ n for some integer α ≥ λ ≥ F , its d. c. is defined as the probability that two randomly chosen elementsin F commute, and Antol´ın, Martino and Ventura have recently generalised this concept to allfinitely generated groups. It has been conjectured that the d. c. of a group G of exponentialgrowth is zero. This paper verifies the conjecture (for certain generating sets) when G is aright-angled Artin group or, more generally, a graph product of groups of rational growth inwhich centralisers of non-trivial elements are “uniformly small”. Introduction
Let G be a group which has a finite generating set X . For any element g ∈ G , let | g | = | g | X bethe word length of g with respect to X . For any n ∈ Z ≥ , let B G,X ( n ) := { g ∈ G | | g | X ≤ n } be the ball in G with respect to X of radius n , and let S G,X ( n ) := { g ∈ G | | g | X = n } be the sphere in G with respect to X of radius n . One writes B G ( n ) or B ( n ) for the ball (and S G ( n ) or S ( n ) for the sphere) if the generating set or the group itself is clear. A group G issaid to have exponential growth if(1) lim inf n →∞ log | B G,X ( n ) | n > subexponential growth otherwise; note that as there are at most (2 | X | ) n words over X ± oflength n , the limit in (1) is finite, so the group cannot have ‘superexponential’ growth. A group G is said to have polynomial growth of degree d if d := lim sup n →∞ log | B G,X ( n ) | log n < ∞ and superpolynomial growth otherwise. It is well-known that having exponential growth orpolynomial growth of degree d is independent of the generating set X .The pairs ( G, X ) as above considered in this paper will have some special properties. In partic-ular, consider the ( spherical ) growth series s G,X ( t ) of a finitely generated group G with a finitegenerating set X , defined by s G,X ( t ) = X g ∈ G t | g | X = ∞ X n =0 | S G,X ( n ) | t n . Mathematics Subject Classification.
Key words and phrases.
Graph products of groups, degree of commutativity, rational growth series.
Cases of particular interest includes pairs (
G, X ) for which s G,X ( t ) is a rational function, i.e.a ratio two polynomials; in this case G is said to have rational growth with respect to X . Ingeneral, this property depends on the chosen generating set: for instance, the higher Heisenberggroup G = H ( Z ) has two finite generating sets X , X such that s G,X ( t ) is rational but s G,X ( t ) is not [19].Rational growth series implies some nice properties on the growth of a group. In particular,one can obtain the first main result of this paper: Theorem 1.
Let G be an infinite group with a finite generating set X such that s G,X ( t ) is arational function. Then there exist constants α ∈ Z ≥ , λ ∈ [1 , ∞ ) and D > C > such that Cn α λ n ≤ | S G,X ( n ) | ≤ Dn α λ n for all n ≥ . Some of the ideas that go into the proof of Theorem 1 appear in the work of Stoll [19], whereasymptotics of ball sizes are used to show that the higher Heisenberg group G = H ( Z ) has afinite generating set X such that the series s G,X ( t ) is transcendental. Remark 2.
It is clear that, with the assumptions and notation as above, Theorem 1 implieslim inf n →∞ | S G,X ( n ) | n α λ n ≥ C > n →∞ | S G,X ( n ) | n α λ n ≤ D < ∞ . It is easy to check that the converse implication is also true. In particular, the conclusion ofTheorem 1 is equivalent to the statement that there exist α ∈ Z ≥ and λ ∈ [1 , ∞ ) such thatlim inf n →∞ | S G,X ( n ) | n α λ n > n →∞ | S G,X ( n ) | n α λ n < ∞ . Theorem 1 agrees with the result for hyperbolic groups. Indeed, it is known that if G is ahyperbolic group and X is a finite generating set, then s G,X ( t ) is rational [14, Theorem 8.5.N].In this case the Theorem gives a weaker version of [8, Th´eor`eme 7.2], which states that theconclusion of Theorem 1 holds with α = 0.As an application of Theorem 1 a calculation of degree of commutativity is provided. For afinite group F , the degree of commutativity of F was defined by Erd˝os and Tur´an [10] andGustafson [15] as(2) dc( F ) := |{ ( x, y ) ∈ F | [ x, y ] = 1 }|| F | , i.e. the probability that two elements of F chosen uniformly at random commute. In [1], Antol´ın,Martino and Ventura generalise this definition to infinite finitely generated groups: Definition 3.
Let G be a finitely generated group and let X be a finite generating set for G .The degree of commutativity for G with respect to X isdc X ( G ) := lim sup n →∞ |{ ( x, y ) ∈ B G,X ( n ) | [ x, y ] = 1 }|| B G,X ( n ) | = lim sup n →∞ P x ∈ B G,X ( n ) | C G ( x ) ∩ B G,X ( n ) || B G,X ( n ) | , where C G ( x ) is the centraliser of x in G .Note that if G is finite then for any generating set X one has B G,X ( N ) = G for all sufficientlylarge N , so this definition agrees with (2).It is known that dc X ( G ) = 0 when G is either a non-virtually-abelian residually finite groupof subexponential growth [1, Theorem 1.3] or a non-elementary hyperbolic group [1, Theorem1.7], independently of the generating set X . It has been conjectured that indeed dc X ( G ) = 0whenever G has superpolynomial growth [1, Conjecture 1.6]. ATIONAL GROWTH AND DEGREE OF COMMUTATIVITY OF GRAPH PRODUCTS 3
The interest of this paper is the degree of commutativity of graph products of groups.
Definition 4.
Let Γ be a finite simple (undirected) graph, and let H : V (Γ) → G be a map fromthe vertex set of Γ to the category G of groups; suppose that H ( v ) ≇ { } for each v ∈ V (Γ).Let ˜ G (Γ , H ) := ∗ v ∈ V (Γ) H ( v )be a free product of groups, and let R (Γ , H ) := { [ g, h ] | g ∈ H ( v ) , h ∈ H ( w ) , { v, w } ∈ E (Γ) } . Then the graph product associated with Γ and H is defined to be the group G (Γ , H ) := ˜ G (Γ , H ) / hh R (Γ , H ) ii ˜ G (Γ , H ) . In particular, this is the construction of right-angled Artin (respectively
Coxeter ) groups if H ( v ) ∼ = Z (respectively H ( v ) ∼ = C ) for all v ∈ Γ.This paper considers groups G which, together with their finite generating sets X , belong to acertain class, defined as follows. Definition 5.
Say a pair (
G, X ) with a group G and a finite generating set X of G is a rationalpair with small centralisers if the following two conditions hold:(i) s G,X ( t ) is a rational;(ii) there exist constants P, β ∈ Z ≥ such that | C G ( g ) ∩ B G,X ( n ) | ≤ P n β for all n ≥ g ∈ G .Note that condition (ii) is independent of the choice of a generating set X : indeed, as anyword metrics on G associated with generating sets X and ˆ X are bi-Lipschitz equivalent, theinequality | C G ( g ) ∩ B G,X ( n ) | ≤ P n β implies the inequality | C G ( g ) ∩ B G, ˆ X ( n ) | ≤ ˆ P n β for someˆ P ∈ Z ≥ depending only on ˆ X and P .It was shown in [7] that, given a finite simple graph Γ with a group H ( v ) and a finite generatingset X ( v ) ⊆ H ( v ) associated to every vertex v ∈ V (Γ), if s H ( v ) ,X ( v ) ( t ) is rational for each v ∈ V (Γ) then so is s G (Γ , H ) ,X (Γ , H ) ( t ), where X (Γ , H ) = F v ∈ V (Γ) X ( v ).If G (Γ , H ) has exponential growth, then, together with an explicit form of centralisers in G (Γ , H ), described in [2], Theorem 1 can be used to compute the degree of commutativityof G (Γ , H ): Theorem 6.
Let Γ be a finite simple graph, and for each vertex v ∈ V (Γ) , let ( H ( v ) , X ( v )) bea rational pair with small centralisers. Suppose that G (Γ , H ) has exponential growth, and let X = F v ∈ V (Γ) X ( v ) . Then dc X ( G (Γ , H )) = 0 . Remark 7.
Theorem 6 is enough to confirm [1, Conjecture 1.6] in this setting: that is, either G = G (Γ , H ) is virtually abelian, or dc X ( G ) = 0. Indeed, G (Γ , H ) has subexponential growthif and only if all the H ( v ) have subexponential growth, the complement Γ C of Γ contains nolength 2 paths, and H ( v ) ∼ = C for every non-isolated vertex v of Γ C . In this case, rationalityof s H ( v ) ,X ( v ) ( t ) implies that the H ( v ) all have polynomial growth (by Theorem 1, for instance).Thus G (Γ , H ) is a direct product of groups of polynomial growth: namely, the group H ( v ) foreach isolated vertex v of Γ C , and an infinite dihedral group for each edge in Γ C . Consequently, G (Γ , H ) itself has polynomial growth, and so [1, Corollary 1.5] implies that either G (Γ , H ) isvirtually abelian, or dc X ( G (Γ , H )) = 0.Cases of particular interest of Theorem 6 include right-angled Artin groups and graph productsof finite groups. More generally, let us note two special cases of pairs of ( G, X ) satisfyingDefinition 5:
MOTIEJUS VALIUNAS (i) Let G be virtually nilpotent, and X be a finite generating set with s G,X ( t ) rational: inparticular, this holds whenever G is virtually abelian [3] and for G = H , the integralHeisenberg group [9]. It was shown that by Wolf [20] that if G is virtually nilpotent thenit has polynomial growth (by Gromov’s Theorem [13], the converse is also true), and sopart (ii) of Definition 5 holds trivially by bounding growth of centralisers by the growthof G itself.(ii) Let G be a torsion-free hyperbolic group, and X be any finite generating set. Cannon [6]and Gromov [14, Theorem 8.5.N] have shown that hyperbolic groups have rational growthwith respect to any generating set, and all infinite-order elements have virtually cycliccentralisers. Moreover, for any torsion-free hyperbolic group G with a finite generatingset X , there is a constant P > | C G ( g ) ∩ B G,X ( n ) | ≤ P n for all n ≥ g ∈ G : see the proof of Theorem 1.7 in [1] for details and references.The paper is structured as follows. Section 2 applies to all infinite groups with rational sphericalgrowth series and is dedicated to a proof of Theorem 1. Section 3 is used to prove Theorem 6. Acknowledgements.
The author would like to give special thanks to his Ph.D. supervisor,Armando Martino, without whose help and guidance this paper would not have been possible. Hewould also like to thank Yago Antol´ın, Charles Cox and Enric Ventura for valuable discussionsand advice, as well as Ashot Minasyan and anonymous referees for their comments on thismanuscript. Finally, the author would like to give credit to Gerald Williams for a questionwhich led to generalising a previous version of Theorem 6. The author was funded by EPSRCStudentship 1807335. Groups with rational growth series
This section provides a proof of Theorem 1. Let G be an infinite group, and suppose that thegrowth series of G with respect to a finite generating set X is a rational function. In particular,the spherical growth series is s ( t ) = s G,X ( t ) = ∞ X n =0 S ( n ) t n = p ( t ) q ( t )where S ( n ) = S G ( n ) = S G,X ( n ) := | S G,X ( n ) | , and q ( t ) = q t c r Y i =1 (1 − λ i t ) α i +1 and p ( t ) = p t ˜ c ˜ r Y i =1 (1 − ˜ λ i t ) ˜ α i +1 are non-zero polynomials with no common roots (and so either c = 0 or ˜ c = 0), with α i , ˜ α i ∈ Z ≥ for all i . Since the series ( S ( n )) ∞ n =0 grows at most exponentially, s ( t ) is analytic (and socontinuous) at 0, hence one has1 = S (0) = lim t → s ( t ) = p q lim t → t ˜ c − c and so c = ˜ c and p = q . Thus c = ˜ c = 0 and, without loss of generality, q = p = 1.Coefficients of such a series are described in [16, Lemma 1]; in particular, it follows that(3) S ( n ) = r X i =1 α i X j =0 b i,j n j λ ni for n large enough, with b i,α i = 0 for all i .Now consider the terms of (3) that give a non-negligible contribution to S ( n ) for large n . Inparticular, one may assume without loss of generality that λ := | λ | = | λ | = · · · = | λ ˜ k | > | λ ˜ k +1 | ≥ | λ ˜ k +2 | ≥ · · · ≥ | λ r | ATIONAL GROWTH AND DEGREE OF COMMUTATIVITY OF GRAPH PRODUCTS 5 for some ˜ k ≤ r and that α := α = α = · · · = α k > α k +1 ≥ α k +2 ≥ · · · ≥ α ˜ k for some k ≤ ˜ k . Note that one must have λ ≥
1: otherwise the radius of convergence of s ( t ) is λ − > P n S ( n ) converges, contradicting the fact that G is infinite.For n ∈ Z ≥ , define c n = k X j =1 b j,α exp( iϕ j n )where λ j = λ exp( iϕ j ) for some ϕ j ∈ ( − π, π ], for 1 ≤ j ≤ k . It follows that(4) S ( n ) = n α λ n ( c n + o (1))as n → ∞ . In particular, since S ( n ) ∈ (0 , ∞ ) ⊆ R for all n , it follows that(5) lim inf n →∞ Re( c n ) ≥ n →∞ Im( c n ) = 0 . It is clear that lim sup n →∞ S ( n ) n α λ n ≤ k X j =1 | b j,α | , which shows existence of the constant D in Theorem 1; in order to prove the Proposition, it isenough to show that lim inf n →∞ S ( n ) / ( n α λ n ) >
0. However, this bound does not follow solelyfrom the fact that s ( t ) is a rational function: see Example 12 (i) at the end of this section. Remark 8.
Clearly, for any n , n ≥
0, if g ∈ G has | g | X = n + n (respectively | g | X ≤ n + n ),then one can write g = g g where | g j | X = n j (respectively | g j | X ≤ n j ) for j ∈ { , } . Thisgives injections S ( n + n ) → S ( n ) × S ( n ) and B ( n + n ) → B ( n ) × B ( n ) by mapping g ( g , g ). In particular, it follows that S ( n + n ) ≤ S ( n ) S ( n ) and B ( n + n ) ≤ B ( n ) B ( n )for any n , n ∈ Z ≥ . This property is called submultiplicativity of sphere and ball sizes in G .The aim is now to show that submultiplicativity of the sequence ( S ( n )) ∞ n =0 , together withrationality of s ( t ), implies the conclusion of Theorem 1. As the b j,α are non-zero and the ϕ j aredistinct, given (5) the following result seems highly likely: Lemma 9.
The numbers c n are real, and for some constant δ > , the set E δ := { n ∈ Z ≥ | c n ≥ δ } is relatively dense in [0 , ∞ ) , i.e. the inclusion E δ ֒ → [0 , ∞ ) is a (1 , K ) -quasi-isometry for some K ≥ . However, the author has been unable to come up with a straightforward proof of Lemma 9without using some additional theory on ‘quasi-periodicity’ of the sequence ( c n ) ∞ n =0 . Beforegiving a proof, let us deduce Theorem 1 from Lemma 9.Assuming Lemma 9, one can find N ∈ Z ≥ such that for all n , there exists a β = β n ∈ { , . . . , N } with c n + β ≥ δ . Define R := max { λ − β S ( β ) | ≤ β ≤ N } , and let M ∈ Z ≥ be such that for all n ≥ M , one has S ( n ) ≥ n α λ n (cid:18) c n − δ (cid:19) MOTIEJUS VALIUNAS (such an M exists by (4)). Then submultiplicativity of sphere sizes implies that for all n ≥ M , δ n + β n ) α λ n + β n ≤ (cid:18) c n + β n − δ (cid:19) ( n + β n ) α λ n + β n ≤ S ( n + β n ) ≤ S ( n ) S ( β n ) ≤ S ( n ) Rλ β n . It follows that S ( n ) ≥ δ R ( n + β n ) α λ n ≥ δ R n α λ n for n ≥ M , showing that lim inf n →∞ S ( n ) n α λ n ≥ δ R > , which shows existence of the constant C >
Proof of Lemma 9.
To prove the Lemma, one may employ a digression into a certain classof functions from R to C , called ‘uniformly almost periodic functions’. The theory for thesefunctions is presented in a book by Besicovitch [5].Let f : R → C be a function. Given ε >
0, define the set E ( f, ε ) ⊆ R to be the set of allnumbers τ ∈ R (called the translation numbers for f belonging to ε ) such thatsup x ∈ R | f ( x + τ ) − f ( x ) | ≤ ε. The function f is said to be uniformly almost periodic ( u. a. p. ) if, for any ε >
0, the set E ( f, ε ) is relatively dense in R , i.e. the inclusion E ( f, ε ) ֒ → R is a (1 , K )-quasi-isometry forsome K ≥
0. It is easy to see that any periodic function is u. a. p., and that every continuousu. a. p. function is bounded.Now note that the function c : R → C t k X j =1 b j,α exp( iϕ j t )is a sum of continuous periodic functions, and so is a continuous u. a. p. function by [5, Section1.1, Theorem 12]. By definition, c n = c ( n ) for any n ∈ Z ≥ .The aim is to show that the function ¯ c : t c ( ⌊ t ⌋ ) is also u. a. p. For this, note that c iseverywhere differentiable and the derivative c ′ ( t ) is a sum of continuous periodic functions, so iscontinuous and u. a. p. – in particular, it is bounded, by some R >
0, say. For a given ε ∈ (0 , R ),set a constant M := ε/ (cid:0) (cid:0) πε R (cid:1)(cid:1) and define f : R → R by f ( t ) = M sin( πt ). It is easy tocheck that(6) E (cid:16) f, ε (cid:17) ⊆ [ n ∈ Z h n − ε R , n + ε R i . For any τ ∈ R , define n τ = (cid:4) τ + (cid:5) ∈ Z to be the nearest integer to τ . Pick τ ∈ E (cid:0) f, ε (cid:1) ∩ E (cid:0) c, ε (cid:1) – then | c ( x + τ ) − c ( x ) | ≤ ε for all x ∈ R , and, by (6), | τ − n τ | ≤ ε R , so in particular | c ( x + τ ) − c ( x + n τ ) | ≤ ε for all x ∈ R by the choice of R . Thus | c ( x + n τ ) − c ( x ) | ≤ ε for all x ∈ R , i.e. n τ ∈ E ( c, ε ).But by [5, Section 1.1, Theorem 11], the set E (cid:0) f, ε (cid:1) ∩ E (cid:0) c, ε (cid:1) is relatively dense, hence (by theprevious paragraph) so is the set E ( c, ε ) ∩ Z . However, for any n ∈ E ( c, ε ) ∩ Z and any x ∈ R one has | ¯ c ( x + n ) − ¯ c ( x ) | = | c ( ⌊ x + n ⌋ ) − c ( ⌊ x ⌋ ) | = | c ( ⌊ x ⌋ + n ) − c ( ⌊ x ⌋ ) | ≤ ε and so E ( c, ε ) ∩ Z ⊆ E (¯ c, ε ) ∩ Z . It follows that E (¯ c, ε ) ∩ Z is relatively dense (and so thefunction ¯ c : t c ( ⌊ t ⌋ ) is u. a. p.). ATIONAL GROWTH AND DEGREE OF COMMUTATIVITY OF GRAPH PRODUCTS 7
Now recall that (5) provides constraints for limits of sequences (Re( c n )) and (Im( c n )): namely,(7) lim inf n →∞ Re( c n ) ≥ n →∞ Im( c n ) = 0 . It is easy to see that c n ∈ R ≥ for all n : indeed, if either Re( c n ) = − δ < | Im( c n ) | = δ > n then the fact that the set E (¯ c, δ/ ∩ Z is relatively dense contradicts (7). Similarly,if c N > N then the set E (¯ c, δ ) ∩ Z is a relatively dense set contained in the set { n ∈ Z | c ( n ) ≥ δ } , where δ = c N /
2. To prove Lemma 9 it is therefore enough to show that thesequence ( c n ) ∞ n =0 is not identically zero.Now recall that the sequence ( c n ) is defined by c n = k X j =1 b j,α exp( iϕ j n ) , and suppose for contradiction that c n = 0 for all n ∈ Z ≥ , and in particular for 0 ≤ n ≤ k − M v = 0, where M = · · · iϕ ) exp( iϕ ) · · · exp( iϕ k )... ... . . . ...exp( iϕ ) k − exp( iϕ ) k − · · · exp( iϕ k ) k − and v = b ,α − b ,α − ... b k,α − . Thus M has a zero eigenvalue and so det M = 0. But M t is a Vandermonde matrix withpairwise distinct rows, so det M = 0. This gives a contradiction which completes the proof. (cid:3) Remark 10.
A stronger conclusion of Theorem 1 holds if in addition s G,X ( t ) is a positive rational function, i.e. it is contained in the smallest sub-semiring of C ( t ) containing the semiring Z ≥ [ t ] and closed under quasi-inversion, f ( t ) (1 − f ( t )) − (for f ( t ) ∈ C ( t ) with f (0) = 0).This is the case in particular if there exists a language L in ( X ∪ X − ) ∗ that is regular (i.e.recognised by a finite state automaton), the monoid homomorphism Φ : L → G extending theinclusion X ∪ X − ֒ → G is a bijection, and L consists only of geodesic words in the Cayley graphof G with respect to X , i.e. the length of any word l ∈ L is | Φ( l ) | X . If s G,X ( t ) is a positiverational function, then the numbers ϕ j above are in fact rational multiples of π [4], and as aconsequence the sequence ( c n ) is periodic.However, the author has not been able to find a reason why the function s G,X ( t ), in case it isrational, must also be positive. In particular, one can find pairs ( G, X ) such that s G,X ( t ) isrational but there are no regular languages L as above, and one can even find groups G suchthat this holds for ( G, X ) for any generating set X . For instance, it can be shown that growthof the 2-step nilpotent Heisenberg group G = H = h a, b, c | [ a, b ] = c, [ a, c ] = [ b, c ] = 1 i is rational with respect to any generating set [9, Theorem 1], but there are no languages L asabove when G is a 2-step nilpotent group that is not virtually abelian [18, Corollary 3].It is easy to check that the conclusion of Theorem 1 implies that(8) lim inf n →∞ | B G,X ( n ) | n ˆ α λ n > n →∞ | B G,X ( n ) | n ˆ α λ n < ∞ , where ˆ α = α + 1 if λ = 1 and ˆ α = α otherwise. Asymptotics similar to these have been obtainedfor nilpotent groups, even without the condition on rational growth. In particular, in [17] Pansushowed that given a nilpotent group G with a finite generating set X , there exists ˆ α ∈ Z ≥ such MOTIEJUS VALIUNAS that | B G,X ( n ) | n ˆ α → C as n → ∞ for some C >
0. Moreover, in [19] Stoll calculates the constant C for certain 2-step nilpotent groups G explicitly to show that the corresponding growth series s G,X ( t ) cannot be rational. However, in general – for groups that are not virtually nilpotent –one cannot expect lim sup and lim inf in (8) to be equal, as the hyperbolic group C ∗ C shows:see [12, § Theorem 11.
Let ( a n ) ∞ n =0 be a submultiplicative sequence of numbers in Z ≥ such that s ( t ) = P a n t n is a rational function. Then there exist constants α ∈ Z ≥ , λ ∈ [1 , ∞ ) and D > C > such that for all n ≥ , Cn α λ n ≤ a n ≤ Dn α λ n . The example below shows that both submultiplicativity and rationality are necessary require-ments.
Example 12. (i) Let p ( t ) = 1 + 12 t − t and q ( t ) = (1 − t )(1 − t )(1 − ωt )(1 − ωt ) , where ω is a th primitive root of unity. Let s ( t ) , ( a n ) , λ , α and ( c n ) be as above. Then λ = 2 and α = 0 , and [16, Lemma 1] can be used to calculate a n = c n n + 1 where c n = 4 − ω n − ω n = , n ≡ , , n ≡ ± , , n ≡ ± , , n ≡ . But as c n = 0 for infinitely many values of n , one has lim inf n →∞ a n / ( n α λ n ) = 0 . Note that in this case a = 257 > a a , so the sequence ( a n ) is not submultiplicative.(ii) For n ≥ , let a n = 2 b ( n ) , where b ( n ) is the sum of digits in the binary representation of n . Then ( a n ) is a submultiplicative sequence, but P a n t n is not a rational function. Foreach n ≥ , one has a n − = 2 n and a n = 2 . Thus lim inf n →∞ a n n ≤ lim inf n →∞ n = 0 and lim sup n →∞ a n ≥ lim sup n →∞ n = ∞ , so ( a n ) does not satisfy the conclusion of Theorem 11 for any λ ≥ and α ∈ Z ≥ . Degree of commutativity
The aim of this section is to prove Theorem 6. For this, let Γ be a finite simple graph andfor each v ∈ V (Γ), let ( H ( v ) , X ( v )) be a rational pair with small centralisers (see Definition5). To simplify notation, suppose in addition that the sets X ( v ) are symmetric and do not contain the identity 1 ∈ H ( v ): clearly this does not affect the results. Suppose in addition that G = G (Γ , H ) is a group of exponential growth. One thus aims to show that dc X ( G ) = 0, where X = F v ∈ V (Γ) X ( v ). ATIONAL GROWTH AND DEGREE OF COMMUTATIVITY OF GRAPH PRODUCTS 9
Preliminaries.
This subsection collects the terminology and preliminary results used inthe proof of Theorem 6.Let ℓ n : X ∗ → Z ≥ be the normal form length function ( n in ℓ n stands for ‘normal’): for w ∈ X ∗ ,set ℓ n ( w ) := m where m is the minimal integer for which w ≡ w w · · · w m as words, where w i ∈ X ( v i ) ∗ for some v i ∈ V (Γ). Moreover, let ℓ w : X ∗ → Z ≥ be the word length function ( w in ℓ w stands for ‘word’), i.e. let ℓ w ( w ) be the number of letters in w ∈ X ∗ .The following result says that given any word w ∈ X ∗ representing g ∈ G , there is a simplealgorithm to transform it into a word ˆ w representing g with ℓ n ( ˆ w ) or ℓ w ( ˆ w ) small. This followsquite easily from a result of Green [11]. Proposition 13.
Let ℓ : X ∗ → Z ≥ be either ℓ = ℓ n or ℓ = ℓ w . Let w ∈ X ∗ be a wordrepresenting an element g ∈ G , and let ˆ w be a word representing g with ( ℓ ( ˆ w ) , ℓ w ( ˆ w )) minimal(in the lexicographical ordering) among such words. Then ˆ w can be obtained from w by applyinga sequence of moves of two types:(i) for some w u ∈ X ( u ) ∗ and w v ∈ X ( v ) ∗ with { u, v } ∈ E (Γ) , replacing a subword w u w v with w v w u ;(ii) for some v ∈ V (Γ) and some subword w ∈ X ( v ) ∗ , replacing the subword w with a word w ∈ X ( v ) ∗ representing the same element in H ( v ) , such that ℓ w ( w ) ≤ ℓ w ( w ) .Proof. Suppose first that ℓ = ℓ n , and let ˆ w ≡ w · · · w m , where w i ∈ X ( v i ) ∗ for some v i ∈ V (Γ)and m = ℓ n ( w ). In [11, Theorem 3.9], Green showed that by using moves (i) and (ii) we cantransform w into a word ˆ w ′ ≡ w ′ · · · w ′ m where w ′ i ∈ X ( v i ) ∗ and w i , w ′ i represent the sameelement of H ( v ). Notice that we have ℓ w ( w i ) ≤ ℓ w ( w ′ i ) for each i : otherwise, existence of theword w · · · w i − w ′ i w i +1 · · · w m would contradict the minimality of ˆ w . Thus a sequence of moves(ii) allows us to transform ˆ w ′ into ˆ w , as required.Suppose now that ℓ = ℓ w . Let ˆ w n ∈ X ∗ be a word representing g with ( ℓ n ( ˆ w n ) , ℓ w ( ˆ w n )) minimalamong all such words. Then the result for ℓ = ℓ n says that ˆ w can be transformed into ˆ w n byusing the moves (i)–(ii). Notice that if w ′ ∈ X ∗ is obtained from w ∈ X ∗ by applying move (i)or (ii), then ℓ w ( w ′ ) ≤ ℓ w ( w ), and if the equality holds then there exists a move that transforms w ′ back into w . By definition of ˆ w , no moves strictly decreasing the word length are usedwhen transforming ˆ w to ˆ w n , and so there exists a sequence of moves transforming ˆ w n into ˆ w aswell. Thus we may apply moves (i)–(ii) to obtain ˆ w n from w and subsequently ˆ w from ˆ w n , asrequired. (cid:3) Note that it follows from the proof of Proposition 13 that minimal values of ℓ n ( w ) and ℓ w ( w )can be obtained simultaneously. This justifies the following: Definition 14.
For g ∈ G , define a normal form of g to be a word w ∈ X ∗ with both ℓ n ( w )and ℓ w ( w ) minimal (so that ℓ w ( w ) = | g | X ). Write w = w w · · · w n for w i ∈ X , and define the support of g as supp( g ) := { v ∈ V (Γ) | w i ∈ X ( v ) for some i } ;by Proposition 13 this does not depend on the choice of w .Now suppose for contradiction that dc X ( G ) >
0. That means that for some constant ε > X g ∈ B ( n ) | C G ( g ) ∩ B ( n ) | B ( n ) ≥ ε for infinitely many values of n , where C G ( g ) denotes the centraliser of an element g ∈ G , and B ( n ) = B G ( n ) = B G,X ( n ) := | B G,X ( n ) | .In the proof certain conjugates of elements in G will be considered. In particular, let g ∈ G ,and pick a conjugate ˜ g ∈ G of g such that g = p − g ˜ gp g with | g | = 2 | p g | + | ˜ g | and such that | ˜ g | is minimal subject to this. If p g = 1, then g is called cyclically reduced ; hence ˜ g is cyclicallyreduced. Note that being cyclically reduced is a weaker condition than being cyclically normalin the sense of [2].For any subset A ⊆ V (Γ), let G A denote G (Γ( A ) , H | A ), where Γ( A ) is the full subgraph of Γspanned by A . These will be viewed as subgroups (called the special subgroups ) of G . One mayalso define the link of A to belink A = { u ∈ V (Γ) | ( u, v ) ∈ E (Γ) for all v ∈ A } . Before carrying on with the proof, consider the sequence ( d n ) ∞ n =0 where d n := |{ ( x, y ) ∈ B G,X ( n ) | [ x, y ] = 1 }| B G,X ( n ) . One aims to show that d n → n → ∞ . Note that for many groups of exponential growth,including all the non-elementary hyperbolic groups [1], the sequence ( d n ) ∞ n =0 converges to zeroexponentially fast. However, the following example shows that this is not always the case forgraph products. The result of Theorem 6 may be therefore more delicate than one might think. Example 15.
Suppose Γ is a complete bipartite graph K k,k , i.e. Γ has vertex set V (Γ) = { u , . . . , u k , v , . . . , v k } and edge set E (Γ) = {{ u i , v j } | ≤ i, j ≤ k } , and let H ( u ) ∼ = Z with generators X ( u ) = { x u , x − u } for each u ∈ V (Γ) . In this case one has G (Γ , H ) ∼ = F k × F k (direct product of two free groups of rank k ) and so one can calculate spheresizes in G (Γ , H ) and its special subgroups easily. Note that clearly (by the definition of link)every element of G A ≤ G commutes with every element of G link A ≤ G . Now consider the casewhere A = { u , . . . , u k } and so link A = { v , . . . , v k } . It follows that { ( x, y ) ∈ B ( n ) | [ x, y ] = 1 } ⊇ B G A ( n ) × B G link A ( n ) . An explicit computation shows that B G A ( n ) = B G link A ( n ) = k (2 k − n − k − and B G ( n ) = 2 k n (2 k − n ( k − k −
1) + e (2 k − n + e where e = e ( k ) and e = e ( k ) are some constants. It follows that d n ≥ B G A ( n ) B G link A ( n ) B G ( n ) ∼ (cid:18) k − kn (cid:19) as n → ∞ . In particular, the sequence ( d n ) ∞ n =0 converges to zero only at a polynomial rate for G = G (Γ , H ) . The proof of Theorem 6 is based on the fact that if (9) held for infinitely many n then therewould exist a subset A ⊆ V (Γ) such that the growth of both G A and G link A would be comparableto that of G . More precisely, the outline of the proof is as follows:(i) finding such a subset A ⊆ V (Γ) and showing that G A is not negligible in G , i.e. B GA ( n ) B G ( n ) n → ∞ (subsection 3.2);(ii) finding a collection H of subgroups of G having (uniformly) polynomial growth such that,for all H ∈ H , G link A × H is a subgroup of G and | ( G link A × H ) ∩ B G ( n ) | B G ( n ) is uniformly boundedbelow as n → ∞ (subsection 3.3);(iii) using the embedding G A × G link A ⊆ G and Theorem 1 to obtain a contradiction (subsection3.4). ATIONAL GROWTH AND DEGREE OF COMMUTATIVITY OF GRAPH PRODUCTS 11
A non-negligible special subgroup.
Note that (9) can be rewritten as(10) X A ⊆ V (Γ) X g ∈ B ( n )supp(˜ g )= A | C G ( g ) ∩ B ( n ) | B ( n ) ≥ ε and so (10) holds for infinitely many n . But as Γ is finite, there are only 2 | V (Γ) | < ∞ subsets of V (Γ), thus in particular there exists a subset A ⊆ V (Γ) such that(11) X g ∈ B ( n )supp(˜ g )= A | C G ( g ) ∩ B ( n ) | B ( n ) ≥ −| V (Γ) | ε holds for infinitely many n . One may restrict the subset of elements g ∈ G considered evenfurther: Lemma 16.
There exist constants ˜ ε > and s ∈ Z ≥ such that X g ∈ B ( n )supp(˜ g )= A | p g |≤ s | C G ( g ) ∩ B ( n ) | B ( n ) ≥ ˜ ε for infinitely many n .Proof. As G has rational spherical growth series by [7], Theorem 1 says that there exist constants α ∈ Z ≥ , λ ≥ C = C G > D = D G > C such that(12) Cn α λ n ≤ S ( n ) ≤ Dn α λ n for all n ≥
1. As it is also assumed that G has exponential growth, one has λ >
1. It is easy toshow that in this case(13) Cn α λ n < B ( n ) < Dλλ − n α λ n for all n ≥ g ∈ G with | p g | large (even without requiring supp(˜ g ) = A ). Indeed, as any g ∈ G can be written as g = p − g ˜ gp g with | g | = 2 | p g | + | ˜ g | , (12) and (13) imply1 B ( n ) X g ∈ B ( n ) | p g | >s | C G ( g ) ∩ B ( n ) | B ( n ) ≤ |{ g ∈ B ( n ) | | p g | > s }| B ( n ) ≤ ⌊ n ⌋ X i = s +1 S ( i ) B ( n − i ) B ( n ) ≤ DC (cid:18) (cid:19) α λ − n + D λC ( λ − ⌊ n − ⌋ X i = s +1 (cid:18) i ( n − i ) n (cid:19) α λ − i . (14)The first term of the sum above clearly tends to zero as n → ∞ , and the second term is boundedabove by the infinite sum P ∞ i = s +1 i α λ − i , which tends to zero as s → ∞ since the series P i i α λ − i converges. Hence there exists a value of s ∈ Z ≥ which ensures that the right hand side in (14)is less than 2 −| V (Γ) |− ε for n large enough. This means that X g ∈ B ( n )supp(˜ g )= A | p g |≤ s | C G ( g ) ∩ B ( n ) | B ( n ) ≥ −| V (Γ) |− ε for infinitely many n , so setting ˜ ε := 2 −| V (Γ) |− ε completes the proof. (cid:3) Now note that one may write X g ∈ B ( n )supp(˜ g )= A | p g |≤ s | C G ( g ) ∩ B ( n ) | B ( n ) ≤ |{ g ∈ B ( n ) | supp(˜ g ) = A, | p g | ≤ s }| B ( n ) × max (cid:26) | C G ( g ) ∩ B ( n ) | B ( n ) (cid:12)(cid:12)(cid:12)(cid:12) g ∈ B ( n ) , supp(˜ g ) = A, | p g | ≤ s (cid:27) where both terms in the product are bounded above by 1. It follows by Lemma 16 that both( ∗ ) |{ g ∈ B ( n ) | supp(˜ g ) = A, | p g | ≤ s }| B ( n ) ≥ ˜ ε and( † ) max (cid:26) | C G ( g ) ∩ B ( n ) | B ( n ) (cid:12)(cid:12)(cid:12)(cid:12) g ∈ B ( n ) , supp(˜ g ) = A, | p g | ≤ s (cid:27) ≥ ˜ ε hold for infinitely many n .The aim is now to show that ( ∗ ) and ( † ) imply that the special subgroups G A and G link A (re-spectively) are non-negligible in G . For the latter, one may consider explicit forms of centralisersof G : see the next subsection. For the former, note that the set in the numerator consists ofelements g ∈ B G ( n ) which have an expression g = p − g ˜ gp g with p g ∈ B G ( s ) and ˜ g ∈ B G A ( n ). Itfollows that |{ g ∈ B ( n ) | supp(˜ g ) = A, | p g | ≤ s }| ≤ B G ( s ) B G A ( n )and so ( ∗ ) implies that( ∗∗ ) ˜ ε B G ( s ) ≤ B G A ( n ) B G ( n ) ≤ n , where the second inequality comes from the fact that B G A ( n ) ⊆ B G ( n ).3.3. Centralisers in G . In order to use ( † ), one needs to consider forms of centralisers ofelements g ∈ G with supp(˜ g ) = A . Fix an element g ∈ G with supp(˜ g ) = A and note that oneclearly has C G ( g ) = p − g C G (˜ g ) p g , so if | p g | ≤ s then one has(15) | C G ( g ) ∩ B ( n ) | ≤ | C G (˜ g ) ∩ B ( n + 2 s ) | . In particular, it follows from ( † ) that for infinitely many n , there exists an element g ∈ B ( n )with supp( e g ) = A and | p g | ≤ s such that( †† ) ˜ ε ≤ | C G (˜ g ) ∩ B G ( n + 2 s ) | B G ( n ) ≤ B G (2 s );here the second inequality comes from the fact that | C G (˜ g ) ∩ B ( n + 2 s ) | ≤ B ( n + 2 s ) ≤ B ( n ) B (2 s ).Now define an element g ∈ G to be cyclically normal (in the sense of [2]) if either ℓ n ( g ) ≤
1, or n := ℓ n ( g ) ≥ w = w · · · w n ∈ X ∗ of g , where w i ∈ X ( v i ) ∗ for some v i ∈ V (Γ), one has v = v n . Then one has Lemma 17.
For any g ∈ G with supp(˜ g ) = A , there exists an element ˜ p g ∈ G A such that ˆ g := ˜ p g ˜ g ˜ p − g is cyclically normal and supp(ˆ g ) = A .Proof. If ℓ n (˜ g ) ≤ p g = 1 does the job. Thus suppose that n := ℓ n (˜ g ) ≥
2. Let E (˜ g ) := { g | w = w · · · w n ∈ X ∗ is a normal form for ˜ g where w i ∈ X ( v i ) ∗ for some v i ∈ V (Γ) , and w n represents g } be a finite subset of G A . By Proposition 13, any two elements in E (˜ g ) commute, and so, forany two distinct elements g ∈ H ( v ) and g ∈ H ( v ) of E (˜ g ), one has v = v . Now define˜ p g := Q g n ∈ E (˜ g ) g n . Then ˜ p g ∈ G A , and following the proof of [2, Lemma 23] one can see ATIONAL GROWTH AND DEGREE OF COMMUTATIVITY OF GRAPH PRODUCTS 13 that ˆ g := ˜ p g ˜ g ˜ p − g is cyclically normal. Since supp(˜ p g ) ⊆ A and supp(˜ g ) = A , it is clear thatsupp(ˆ g ) ⊆ A . It also follows by [2, Lemma 18] that supp(ˆ g ) ∪ supp(˜ p g ) ⊇ A . Thus one onlyneeds to check that supp(˜ p g ) ⊆ supp(ˆ g ).Suppose for contradiction that there exists some v ∈ supp(˜ p g ) \ supp(ˆ g ), and let g v ∈ E (˜ g ) ∩ H ( v )be the (unique) element. It is easy to see that v / ∈ link( A \ { v } ): otherwise any normal formof ˆ g would contain a subword in X ( v ) ∗ representing g v and so v ∈ supp(ˆ g ). Then, followingagain the proof of [2, Lemma 23], one has ˜ n := ℓ n ( g v ˜ gg − v ) ≤ n −
1, with ˜ n = n − g has no normal form w · · · w n , where w i ∈ X ( v i ) ∗ for some v i ∈ V (Γ), with w and w n representing g − v and g v , respectively. Thus, by minimality of | ˜ g | , clearly ˜ n = n −
1; but thiscannot happen by [2, Lemma 18], since by assumption v / ∈ supp(ˆ g ). Hence supp(˜ p g ) ⊆ supp(ˆ g ),as required. (cid:3) The following Proposition describes growth of centralisers in G . Proposition 18.
Let g, ˜ g ∈ G and A ⊆ V (Γ) be as above. Then C G (˜ g ) = H × · · · × H k × G link A for some subgroups H , . . . , H k ≤ G , and the following hold:(i) for any h ∈ H , . . . , h k ∈ H k and c ∈ G link A , | h · · · h k c | X = | h | X + · · · + | h k | X + | c | X ; (ii) there exist constants D , . . . , D k , α , . . . , α k ∈ Z ≥ such that | H i ∩ B G,X ( n ) | ≤ D i n α i for all n ≥ .Furthermore, the number k ∈ Z ≥ , the D i and the α i only depend on A and not on g .Proof. Let A , . . . , A k ⊆ A form a partition of A such that the graphs Γ( A i ) C are precisely theconnected components of the graph Γ( A ) C , where ∆ C denotes the complement of a graph ∆.Let ˜ p g , ˆ g ∈ G A be as in Lemma 17. Then supp(ˆ g ) = A and so ˆ g can be expressed asˆ g = ˆ g · · · ˆ g k where supp(ˆ g i ) = A i .Now suppose without loss of generality that for some m , the sets A i = { v i } are singletons for1 ≤ i ≤ m , and | A i | ≥ m + 1 ≤ i ≤ k . Then Proposition 25, Theorem 32 and Theorem 52in [2] state that the centraliser of ˆ g in G is C G (ˆ g ) = C H ( v ) (ˆ g ) × · · · × C H ( v m ) (ˆ g m ) × h h m +1 i × · · · × h h k i × G link A where h m +1 , . . . , h k ∈ G are some infinite order elements with supp( h i ) = A i (in fact, one hasˆ g i = h β i i for some β i ∈ Z \ { } ).In particular, since ˜ p g ∈ G A , one has ˜ p g = p · · · p k for some p i ∈ G A i . Thus ˜ p − g q i ˜ p g = p − i q i p i for any q i ∈ G A i , and ˜ p − g ( G link A )˜ p g = G link A , hence C G (˜ g ) = ˜ p − g C G (ˆ g )˜ p g = C H ( v ) (˜ g ) × · · · × C H ( v m ) (˜ g m ) × h ˜ g m +1 i × · · · × h ˜ g k i × G link A where ˜ g i := p − i ˆ g i p i for 1 ≤ i ≤ m , and ˜ g i := p − i h i p i for m + 1 ≤ i ≤ k . Hence, by setting H i := C H ( v i ) (˜ g i ) for 1 ≤ i ≤ m and H i := h ˜ g i i ∼ = Z for m + 1 ≤ i ≤ k one obtains the requiredexpression. By construction, k depends only on A (and not on g ).To show (i), it is enough to note that H i ≤ G A i for each i , and that by construction the subsets A i are pairwise disjoint and disjoint from link A . Indeed, then it follows from Proposition 13that if w i (respectively u ) is a normal form for an element h i ∈ G A i (respectively c ∈ G link A ),then w · · · w k u is a normal form for the element h · · · h k c . This implies (i). To show (ii) and the last part of the Proposition, one may consider cases 1 ≤ i ≤ m and m + 1 ≤ i ≤ k separately. For 1 ≤ i ≤ m , note that, as a consequence of Proposition 13, | h | X = | h | X ( v i ) for all h ∈ H i , and therefore | H i ∩ B G,X ( n ) | = | H i ∩ B H ( v i ) ,X ( v i ) ( n ) | for all n ≥ g i = 1 and that ( H ( v i ) , X ( v i )) is a rational pair with smallcentralisers; it also follows that D i , α i do not depend on g . For m + 1 ≤ i ≤ k , it follows fromthe proof of [2, Lemma 37] that since ˆ g i is cyclically normal and since Γ(supp(ˆ g i )) C = Γ( A i ) C isconnected, one has ℓ n (˜ g γi ) ≥ ℓ n (ˆ g γi ) = | γ | ℓ n (ˆ g i ) for all γ ∈ Z . In particular, | ˜ g γi | X ≥ ℓ n (˜ g γi ) ≥ | γ | for any γ ∈ Z and so | H i ∩ B G,X ( n ) | ≤ n + 1 ≤ n for all n ≥
1. Thus taking D i = 3 and α i = 1 shows (ii); independence from g is clear. (cid:3) Products of special subgroups.
To finalise the proof, one employs the following generalresult:
Lemma 19.
Let G be a group with a finite generating set X . Let H, K ≤ G be subgroups suchthat H × K is also a subgroup of G , i.e. the map H × K → G, ( h, k ) hk is an injectivegroup homomorphism. Suppose that there exist constants α H , α K ∈ Z ≥ , λ H , λ K ∈ [1 , ∞ ) and D > C ≥ such that Cn α H λ nH ≤ | H ∩ S G,X ( n ) | ≤ Dn α H λ nH and Cn α K λ nK ≤ | K ∩ S G,X ( n ) | ≤ Dn α K λ nK for all n ≥ . Furthermore, suppose that | hk | X = | h | X + | k | X for all h ∈ H ( n ) , k ∈ K ( n ) , andthat λ H ≥ λ K . If λ H > λ K , then there exists constant e D = e D ( D, α H , α K , λ H , λ K ) > , whichdoes not depend on H or K , such that | ( H × K ) ∩ S G,X ( n ) | ≤ e Dn α H λ nH for all n ≥ . Furthermore, if λ H = λ K and C > , then no such constant e D exists.Proof. Suppose first that λ H > λ K . Clearly it is enough to show thatlim sup n →∞ | ( H × K ) ∩ S G,X ( n ) | n α H λ nH < ∞ . Fix n ≥
1. As | hk | X = | h | X + | k | X for any h ∈ H , k ∈ K , one has | ( H × K ) ∩ S G,X ( n ) | n α H λ nH = 1 n α H λ nH n X i =0 | H ∩ S G,X ( n − i ) | × | K ∩ S G,X ( i ) |≤ D n − X i =1 (cid:18) λ K λ H (cid:19) i (cid:18) n − in (cid:19) α H i α K + (cid:18) λ K λ H (cid:19) n n α K − α H ! . As λ K /λ H <
1, limits of the first and third term above as n → ∞ are D and 0, respectively.The second term can be bounded above by an upper bound for the series D P i ( λ K /λ H ) i i α K ,which converges by the ratio test. Hence indeed lim sup n →∞ | ( H × K ) ∩ S G,X ( n ) | / ( n α H λ nH ) < ∞ ,which implies the result. It is also clear from the inequality above that e D depends only on D , α H , α K , λ H and λ K .Conversely, suppose that C > λ H = λ K =: λ . Let n ≥
20, so that ⌈√ n ⌉ ≤ n/
4. Then | ( H × K ) ∩ S G,X ( n ) | n α H λ n = 1 n α H λ n n X i =0 | H ∩ S G,X ( n − i ) | × | K ∩ S G,X ( i ) |≥ C n − X i =1 (cid:18) n − in (cid:19) α H i α K ≥ C ⌊ n/ ⌋ X i = ⌈√ n ⌉ (cid:18) (cid:19) α H ( √ n ) α K ≥ C − ( α H +2) n αK +1 . In particular, one has | ( H × K ) ∩ S G,X ( n ) | / ( n α H λ nH ) → ∞ as n → ∞ , implying the result. (cid:3) ATIONAL GROWTH AND DEGREE OF COMMUTATIVITY OF GRAPH PRODUCTS 15
Given this Lemma, the proof can be finalised as follows. Recall (see (12) and (13)) that one hasconstants α ∈ Z ≥ , λ > D V (Γ) > C V (Γ) > C V (Γ) n α λ n ≤ S G ( n ) ≤ D V (Γ) n α λ n and C V (Γ) n α λ n < B G ( n ) < D V (Γ) λλ − n α λ n (16)for all n ≥
1. Now ( ∗∗ ) implies that, for infinitely many n ,(17) e C A n α λ n ≤ B G A ( n ) ≤ e D A n α λ n for some e D A > e C A >
0. But as G A has rational growth with respect to F v ∈ A X ( v ), it followsfrom Theorem 1 that in fact, after modifying the constants e D A and e C A if necessary, (17) holdsfor all n ≥
1, and since λ >
1, after further modifying e C A , one has( ∗∗∗ ) e C A n α λ n ≤ S G A ( n ) ≤ e D A n α λ n for all n ≥ †† ) implies that for infinitely many n ≥ s + 1 there exists g ∈ B ( n ) such that e C ′ ( n − s ) α λ n − s ≤ | C G (˜ g ) ∩ B G,X ( n ) | ≤ e D ′ ( n − s ) α λ n − s for some e D ′ > e C ′ >
0. After decreasing the constant e C ′ > n ,(18) e C ′ n α λ n ≤ | C G (˜ g ) ∩ B G,X ( n ) | ≤ e D ′ n α λ n for some g ∈ B ( n ) with supp( e g ) = A and | p g | ≤ s .Note that G link A has rational growth with respect to F v ∈ link A X ( v ) as it is a special subgroupof G , and so by Theorem 1 it follows that, for all n ≥ e C link A n α λ n ≤ S G link A ( n ) ≤ e D link A n α λ n for some e D link A > e C link A > α ∈ Z ≥ , λ ≥ λ , α ) = ( λ, α ). Indeed, as S G link A ( n ) ⊆ S G ( n ), it follows from (16)that either λ < λ or λ = λ and α ≤ α . Let g ∈ G be such that supp( e g ) = A for all n . ByProposition 18, one has an expression C G ( e g ) = H × · · · × H k × G link A . One now applies Lemma 19 k times. In particular, for each i = k, k − , . . . , H := H i +1 × · · · × H k × G link A ,K := H i ( α H , λ H ) := ( ( α , λ ) if λ > , (0 , λ +12 ) if λ = 1 , ( α K , λ K ) := (cid:18) , λ H + 12 (cid:19) ,C := 0 , and D = D i := max { e D i , e D ′ i } . Here e D ′ i > D i n α i ≤ e D ′ i λ nK for each n ≥
1, where D i and α i are as in Proposition18, e D k is such that S G link A ( n ) ≤ e D k n α H λ nH for all n ≥
1, and, for each i = k − , k − , . . . , e D i = e D ( D i +1 , α H , α K , λ H , λ K ) is the constant given by Lemma 19.It then follows that, for all g ∈ G with supp( e g ) = A and | p g | ≤ s ,(20) | C G ( e g ) ∩ S G,X ( n ) | ≤ e Dn α H λ nH for all n ≥
1, where e D = e D ( D , α H , α K , λ H , λ K ) is the constant, independent from g , given byLemma 19. Since λ H >
1, by further increasing e D we may replace S G,X ( n ) with B G,X ( n ) in (20).But by construction, one has either λ H < λ or λ H = λ and α H ≤ α , and so together with (18)this implies that ( λ H , α H ) = ( λ, α ). Thus, by the choice of ( λ H , α H ), one has ( λ , α ) = ( λ, α ),as claimed. In particular, (19) can be rewritten as( ††† ) e C link A n α λ n ≤ S G link A ( n ) ≤ e D link A n α λ n . Finally, note that the group G A ∪ link A = G A × G link A is a special subgroup of G and so one has S G A ∪ link A ( n ) ⊆ S G ( n ). It then follows from ( ∗∗∗ ), ( ††† ) and Lemma 19 that for any e D > S G ( n ) ≥ S G A ∪ link A ( n ) > e Dn α λ n for some n , which contradicts (16). This completes the proof of Theorem 6. References
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