Expected Depth of Random Walks on Groups
aa r X i v : . [ m a t h . G R ] N ov EXPECTED DEPTH OF RANDOM WALKS ON GROUPS
KHALID BOU-RABEE, IOAN MANOLESCU, AND AGLAIA MYROPOLSKA
Abstract.
For G a finitely generated group and g ∈ G , we say g is detected by a normalsubgroup N ⊳ G if g / ∈ N . The depth D G ( g ) of g is the lowest index of a normal, finite indexsubgroup N that detects g . In this paper we study the expected depth, E [ D G ( X n )] , where X n is a random walk on G . We give several criteria that imply that E [ D G ( X n )] −−−−→ n →∞ X k ≥ G : Λ k ] , where Λ k is the intersection of all normal subgroups of index at most k . In particular,the equality holds in the class of all nilpotent groups and in the class of all linear groupssatisfying Kazhdan Property ( T ) . We explain how the right-hand side above appears as anatural limit and also give an example where the convergence does not hold. Introduction
Let G be a finitely generated group. The depth of an element in G encodes how wellapproximated that element is by finite quotients of the group. The goal of this article is tofind the average depth of an element of G . As such, this question is ill-posed, and a moreprecise one is: what is the asymptotic expected depth of a random walk on the Cayley graphof G ? This question arises naturally when quantifying residual finiteness , or in other words,when studying statistics surrounding the depth function .For g ∈ G and N a normal subgroup of G , we say g is detected by N if g / ∈ N (in otherwords, if g is mapped onto a non-trivial element of G/N ). The depth of g is the lowest indexof a normal, finite index subgroup N that detects g . Formally, for g ∈ G , g = e , set D G ( g ) := min {| G/N | : N ⊳ finite index G and g / ∈ N } . For g = e the above definition would produce a depth equal to min ∅ = ∞ . In the context ofrandom walks, this singularity would produce trivial results. To circumvent this triviality, weinstead define D G ( e ) := 0 . With this definition, G is residually finite if and only if D G ( g ) < ∞ for all elements g ∈ G .Let G be residually finite and S be a finite generating set, which will be always consideredsymmetric. Then the residual finiteness growth function is F SG ( n ) = max g ∈ B SG ( n ) D G ( g ) , where B SG ( n ) is the ball of radius n in the Cayley graph Cay(
G, S ) . This notion was introducedin [BR10] and has been studied for various classes of groups; the relevant results to this paperare listed in §2.1. Date : April 1, 2018.K.B. supported in part by NSF grant DMS-1405609, A.M. supported by Swiss NSF grant P2GEP2_162064.I.M. affiliated to NCCR SwissMAP.
While the residual finiteness growth function reflects the largest depth of an element in theball of radius n , the question of interest in this paper is what we can say about the “average”depth of a “uniform” element of the group. If G is discrete but infinite, there is no naturaldefinition of a uniform probability measure on G , hence no good notion of a uniform element.However, we may try to approach the desired “average” by averages of well defined measures.Two approaches come to mind: • For n ≥ , let Z n be a uniform element in B SG ( n ) , and let a n = E [ D G ( Z n )] = 1 | B SG ( n ) | X g ∈ B SG ( n ) D G ( g ) . We could then say that the average depth of an element of G is lim n a n , provided thatthis limit exists. • Alternatively, one may define a random walk ( X n ) n ∈ N on a Cayley graph Cay ( G, S ) of G , starting from the neutral element e , and set b n = E [ D G ( X n )] . Then define the average depth as lim n b n , again under the condition that the limitexist.One expects that for compliant groups, both limits exist and are equal. We will focus on thesecond situation; but will make reference to the first to stress similarities. The exact definitionof ( X n ) n ≥ as well as a discussion on random walks on groups is deferred to §2.2. We mentionhere only that ( X n ) n ≥ is a lazy random walk, that is a process that at every step remainsunchanged with probability / and takes a step otherwise.It is a general fact that for an integer valued non-negative random variable Y , E ( Y ) = X k ≥ P ( Y > k ) . It may therefore be interesting to study P [ D G ( X n ) > k ] for ( X n ) n ≥ as above.For k ≥ , let Λ k be the intersection of all normal subgroups of G of index at most k . (For k = 0 , , set Λ k = G ). Then, for g ∈ G \ { e } , D G ( g ) > k if and only if g ∈ Λ k . Thus E [ D G ( X n )] = X k ≥ P ( X n ∈ Λ k \ { e } ) . (1)As we will see in Corollary 2.5, P ( X n ∈ Λ k \ { e } ) −−−→ n →∞ G : Λ k ] . One may therefore expect that E [ D G ( X n )] −−−→ n →∞ X k ≥ G : Λ k ] , (2)where the factor appears since [ G : Λ ] = [ G : Λ ] = [ G : G ] = 1 . For this reason, we callthe right hand side of the above the presumed limit . However, the convergence above is farfrom obvious. The main goal of this paper is to provide criteria for G under which (2) holds.We will also provide an example where this is not valid.The finiteness of P k ≥ | G :Λ k | in (2) depends on the group G and is related to the intersectiongrowth i G ( k ) = [ G : Λ k ] of G . It follows from Lemma 3.1 and Theorem 3.2 that P k ≥ | G :Λ k | < XPECTED DEPTH OF RANDOM WALKS ON GROUPS 3 ∞ for finitely generated linear groups. Moreover, finitely generated nilpotent groups enjoythis property as it is a classical result that they are linear (see Segal [Seg83, Chapter 5, §B,Theorem 2] or Hall [Hal69, p. 56, Theorem 7.5]).Our two results ensuring (2) are the following. Theorem 1.1.
Let G be a linear group with Kazhdan Property ( T ) . Then lim n →∞ E [ D G ( X n )] = 2 + X k ≥ G : Λ k ] < ∞ for any finite generating set S of G . In particular, lim n →∞ E [ D G ( X n )] is finite for the speciallinear groups SL k ( Z ) with k ≥ . Theorem 1.2.
Let G be a finitely generated nilpotent group. Then lim n →∞ E [ D G ( X n )] = 2 + X k ≥ G : Λ k ] < ∞ for any finite generating set S of G . In §4.1, it will also be shown that the convergence holds whenever the presumed limit isinfinite. Considering these examples, one may think that the convergence in (2) is alwaysvalid. However, in Proposition 4.8, we exhibit a -generated group for which the presumedlimit is finite but lim n →∞ E [ D G ( X n )] = ∞ .Henceforth, when no ambiguity is possible, we drop the index G from the notation D G ( . ) .2. Preliminaries
Depth function and residual finiteness growth.
This short subsection includessome results on the residual finiteness growth function that we will use in the sequel.
Theorem 2.1 ([BR10]) . Let G be a finitely generated nilpotent group with a generating set S . Then F SG ( n ) ≤ C log( n ) h ( G ) , ∀ n ≥ , where h ( G ) is the Hirsch length of G and C = C ( G, S ) is a constant independent of n . The Prime Number Theorem and Hall’s Embedding Theorem play key roles in the proofof Theorem 2.1. In [BRM15], the following is proved using Gauss’s Counting Lemma to helpquantify Mal’cev’s classical proof of residual finiteness of finitely generated linear groups.
Theorem 2.2 ([BRM15]) . Let K be a field. Let G be a finitely generated subgroup of GL ( m, K ) with a generating set S . Then there exists a positive integer b such that F SG ( n ) ≤ Cn b , ∀ n ≥ , where C=C(G,S) is a constant independent of n . The above results bound from above the residual finiteness growth. Conversely, the follow-ing states that there exist groups with arbitrary large residual finiteness growth.
Theorem 2.3 ([BRS16]) . For any function f : N → N , there exists a residually finite group G and a two element generating set S for G , such that F SG ( n ) ≥ f ( n ) for all n ≥ . The proof of Theorem 2.3 in [BRS16] involves an explicit construction of a finitely generatedgroup embedded in an infinite product of finite simple groups.
XPECTED DEPTH OF RANDOM WALKS ON GROUPS 4
Random walks on groups.
Let G be a finitely generated group with a finite symmetricgenerating set S = { s , . . . , s k } , i.e. such that S − = S . A random walk ( X n ) n ≥ on G is aMarkov chain with state space G and such that X = e G and X n +1 = X n · Y n for n ≥ where Y , Y , . . . are independent and uniform in { s , . . . , s k } .If G is finite with | G | = m , one may consider the transition matrix P of the random walk ( X n ) n ≥ on G defined by P ( x, y ) = 1 | S | X s ∈ S { y = xs } , where { y = xs } = 1 if y = xs and otherwise. It is simply the adjacency matrix of the Cayleygraph Cay(
G, S ) , normalized by | S | . The generating set is considered symmetric so as to havean un-oriented Cayley graph, or equivalently to have P symmetric.Let λ ≥ · · · ≥ λ m ≥ − be the eigenvalues of P and x , . . . , x m be a basis oforthonormal eigenvectors of P (such a basis necessarily exists since P is real and symmetric).Let σ be an initial distribution on G seen as a probability vector of dimension m , and let p u = ( m , . . . , m ) be the uniform distribution on G . It is well-known that the distributionof such random walk converges to the uniform distribution whenever the graph is assumedto not be bipartite. For a general convergence statement one considers a lazy random walkinstead; that is a walk with transition matrix L = I + P . The lazy random walk at time n takes a step of the original random walk with probability and stays at the current vertexwith probability . Notice that the eigenvectors of L are x , . . . , x m and the correspondingeigenvalues are all non-negative µ = + λ = 1 > µ = + λ ≥ · · · ≥ µ m = + λ m ≥ . Lemma 2.4.
Let G be a finite group with a finite symmetric generating set S . With the abovenotation || σL n − p u || ≤ µ n . In particular, | σL n ( g ) − m | ≤ µ n for every g ∈ G .Proof. For n ≥ , the matrix σ · L n is a probability distribution and it represents the distribu-tion of the n -the step of the lazy random walk on G that starts at a random vertex selectedaccording to σ .We write σ = α x + α x + · · · + α m x m , with α , . . . , α m ∈ R . Since x , . . . , x m areeigenvectors, we have σL n = α µ n x + α µ n x + · · · + α m µ nm x m . Notice that µ = 1 , x = √ m (1 , . . . , and α = σ · x T = √ m which implies that α x = ( m , . . . , m ) = p u .We deduce that (cid:13)(cid:13) σL n − p u (cid:13)(cid:13) = (cid:13)(cid:13) α µ n x + · · · + α m µ nm x m (cid:13)(cid:13) ≤ max i =2 ,...,m | µ i | n · q α + · · · + α m ≤ µ n · k σ k . In the last line we used the orthonormality of the base x , . . . , x m . Finally, k σ k ≤ P mi =1 σ i =1 , which shows that the above is bounded by µ n as required. (cid:3) Corollary 2.5.
Let G be a finitely generated infinite group with a finite symmetric generatingset S and let N be a normal subgroup of G of finite index. Consider the lazy random walk ( X n ) n ≥ on Cay(
G, S ) . Then (cid:12)(cid:12)(cid:12) P ( X n ∈ N ) − | G : N | (cid:12)(cid:12)(cid:12) ≤ µ n , (3) where µ is the second largest eigenvalue of the transition matrix of ˜ X n , the lazy random walkon Cay(
G/N, S ) induced by ( X n ) n ≥ . Moreover, P ( X n ∈ N \ { e } ) −−−→ n →∞ | G : N | . (4) XPECTED DEPTH OF RANDOM WALKS ON GROUPS 5
Proof.
To prove (3) it suffices to observe that P ( X n ∈ N ) = P ( ˜ X n = e N ) and conclude byLemma 2.4. Let us now show (4).It is a standard fact (see for instance [VSCC93, Thm. VI.3.3, Thm. VI.5.1]) that, since G is infinite, P ( X n = e ) → , as n → ∞ . Moreover, as discussed above, the eigenvalue µ appearing in (3) is strictly smaller than .These two facts, together with (3), imply (4). (cid:3) In the rest of the paper, we will always consider lazy random walks as described above.Straightforward generalisations are possible, such as to random walks with non-uniform sym-metric transition probabilities – that is, walks taking steps according to a finitely supported,symmetric probability on G , with e having a positive probability (which is to say that thewalk has, at any given step, a positive probability of staying at the same place). Certaintransition probabilities with infinite support (but finite first moment) may also be treated,but small complication arise in specific parts of the proof. For the sake of legibility, we limitourselves to the simple framework of uniform probabilities on symmetric generating sets.2.3. Asymptotic density.
For a given infinite group G generated by a finite set S , considerthe so-called asymptotic density (as defined in [BV02]) of a subset X in G defined as follows ρ S ( X ) = lim sup n →∞ | X ∩ B SG ( n ) || B SG ( n ) | . (5)If G satisfies(6) lim n →∞ | B SG ( n + 1) || B SG ( n ) | = 1 then by [BV02] ρ S is left- and right-invariant; and in particular, ρ ( H ) = | G : H | for a finiteindex subgroup H . Moreover, if (6) holds, the lim sup in (5) is actually a limit.Condition (6) holds for groups of polynomial growth (as a consequence of [Pan08]). Thus,for all k ≥ , recalling the random variables X n and Z n defined in the introduction, lim n →∞ P ( X n ∈ Λ k \ { e } ) = lim n →∞ P ( Z n ∈ Λ k \ { e } ) = 1 | G : Λ k | . For groups with exponential growth however condition (6) fails, and the second limit inthe above display does not necessarily exist. We give next an example for G = F { a,b } , thefree group generated by two elements { a, b } and for a normal subgroup N ⊳ F { a,b } . Take S = { a, b, a − , b − } . For g ∈ F { a,b } , let k g k be the word-length of g , that is, the graphdistance from g to e in Cay(F { a,b } , S ) . Set N = { g ∈ F { a,b } : k g k ∈ N } . It is straightforward to check that N is a normal subgroup of F { a,b } of index . However, | N ∩ B F { a,b } ,S ( n ) || B F { a,b } ,S ( n ) | = n +1 − · n − , for n even , n − · n − , for n odd . It is immediate from the above that P ( Z n ∈ N ) does not converge when n → ∞ .The above example, together with Corollary 2.5, explains the choice of the random walk ( X n ) n ≥ rather than of the uniform variables ( Z n ) n ≥ on B SG ( n ) . XPECTED DEPTH OF RANDOM WALKS ON GROUPS 6
Another reason for this choice relates to sampling. Suppose we have sampled an instance ofthe variable Z n for some n ≥ . In order to then obtain a sample of Z n +1 , one needs to restartthe relatively costly process of sampling a uniform point in B SG ( n + 1) . For the random walkhowever, if X n is simulated for some n ≥ , X n +1 is easily obtained by multiplying X n with arandom element in S . This makes the sampling of a sequence ( X , X , . . . ) much easier thanthat of a sequence ( Z , Z , . . . ) . 3. Residual average
We mentioned in the introduction that our goal is to compute the “average” depth of anelement in G . In addition to the two methods proposed above, that is taking the limit of E [ D G ( X n )] or E [ D G ( Z n )] , one may compactify G so that it has a Haar probability measureand take the average depth with respect to it. The natural way to render G compact isby considering its profinite completion , which we will denote by b G . It is a compact group,with a unique uniform Haar measure which we denote by µ . The depth function D G may beextended by continuity to b G , and we call D b G this extension. Then the residual average of G ,denoted by Ave( G ) , is Ave( G ) := Z b G D b G dµ. For details of the profinite completion construction see [Wil98]. For further details of theresidual average construction see [BRM10].
Lemma 3.1.
For any linear group G , Ave( G ) = 2 + ∞ X k =2 | G : Λ k | . (7)Note that Ave( G ) in (7) is equal to the limit in (2). Proof.
Recall the fact that we have conveniently defined Λ = Λ = G and therefore that µ (Λ ) = µ (Λ ) = 1 . The residual average is then Ave( G ) = ∞ X k =1 k · [ µ (Λ k − ) − µ (Λ k )] = X k ≥ k − X ℓ =0 [ µ (Λ k − ) − µ (Λ k )]= X ℓ ≥ X k>ℓ [ µ (Λ k − ) − µ (Λ k )]= X ℓ ≥ µ (Λ ℓ ) We are authorized to change the order of summation in the third equality, since all the termsin the sum are non-negative. In the last equality, we have used the telescoping sum and thefact that µ (Λ k ) → as k → ∞ . The latter convergence is due to G being residually finite.The first two terms in the last sum above are equal to ; for ℓ ≥ , µ (Λ ℓ ) = | G :Λ ℓ | . Thelemma follows immediately. (cid:3) The following theorem, taken from [BRM10], will be necessary when proving Theorem 1.1.
Theorem 3.2 (Theorem 1.4 [BRM10]) . Let Γ be any finitely generated linear group. Thenthe residual average of Γ is finite. XPECTED DEPTH OF RANDOM WALKS ON GROUPS 7
For completeness, we give a proof of the above. The present proof is based on the one in theoriginal paper, with some adjustments meant to correct certain points. The main differencewith the original proof is that here we focus on the connection to intersection growth. Onehas to be especially careful in proving this result, as the residual finiteness growth may varywhen passing to subgroups of finite index (see [BRK12, Example 2.5]).
Proof.
We follow the proof [BRM10, Theorem 1.4], with some changes and expansions. Ac-cording to [BRM10, Proposition 5.2], there exists an infinite representation ρ : Γ → GL ( n, K ) for some n and K/ Q finite. By [BRM10, Lemma 2.5], it suffices to show that the normalresidual average of ρ (Γ) is finite. Set Λ = ρ (Γ) and set S to be the coefficient ring of Λ .For each δ > , from the proof of [BRM10, Proposition 5.1], there exists a normal residualsystem F δ on Λ given by ∆ j = Λ ∩ ker r j , where r j : GL( n, S ) → GL( n, S/ p k j S,j ) , and [Λ : ∆ j ] ≤ [Λ : ∆ j +1 ] ≤ [Λ : ∆ j ] δ . In addition, we have | r j (Λ) | = O j p ℓ j j where(8) ≤ O j < p n j . We also have for constants
N > ( n )! and C > that(9) ℓ j > N + Cjn and ℓ j + Cn < ℓ j +1 ≤ ℓ j + ( C + 1) n . For each i < j , we claim that the largest power of p j that divides [Λ : ∆ i ] is p n j . To seethis claim, note that if p mj divides O i p ℓ i i , since p i , p j are distinct primes, p mj must divide O i ,however O i < p n i and p i < p j . The claim follows. Since [∆ i : ∆ i ∩ ∆ j ] = [Λ : ∆ j ][∆ j : ∆ i ∩ ∆ j ] / [Λ : ∆ i ] = O j p ℓ j j [∆ j : ∆ i ∩ ∆ j ] / [Λ : ∆ i ] , the aforementioned claim implies that [∆ i : ∆ i ∩ ∆ j ] ≥ p ℓ j − n j . Set Λ i := ∩ in =1 ∆ n , then [Λ j − : Λ j ] = O j p ℓ j j [∆ j : Λ j ] / [Λ : Λ j − ] , and [Λ : Λ j − ] divides [Λ : ∆ ] · · · [Λ : ∆ j − ] for which p ( j − n j is the largest power of p j thatappears as a factor. Hence, we obtain [Λ j − : Λ j ] ≥ p ℓ j − ( j − n j . From this, we obtain the important inequality(10) [Λ : Λ k ] ≥ k Y j =1 p ℓ j − ( j − n j . XPECTED DEPTH OF RANDOM WALKS ON GROUPS 8
To employ (10), we need a comparison function. This is where we deviate from the proofin [BRM10]. Define, for g ∈ Λ \ { } , M ( g ) = min { [Λ : ∆ i ] : g / ∈ ∆ i } . Let ˆ M be the unique continuous extension of M to ˆΓ . Then clearly, D Γ ( g ) ≤ M ( g ) , and so Z ˆ D Γ ( g ) dµ ≤ Z ˆ M ( g ) dµ. By studying the partial sums that define R ˆ M ( g ) dµ , we obtain, for any n , n X k =1 [Λ : ∆ k ] µ (Λ k − \ Λ k ) = n X k =1 [Λ : ∆ k ] (cid:18) k − ] − k ] (cid:19) = [Λ : ∆ ][Λ : Λ ] − [Λ : ∆ n ][Λ : Λ n ] + n − X k =1 [Λ : ∆ k +1 ] − [Λ : ∆ k ][Λ : Λ k ] < [Λ : ∆ ][Λ : Λ ] + n X k =1 [Λ : ∆ k ] δ [Λ : Λ k ] . The last inequality follows from the conclusion of [BRM10, Proposition 5.1] that [Λ : ∆ k +1 ] ≤ [Λ : ∆ k ] δ . Applying (8) and (10) while plugging in the value for [Λ : Λ k ] yields n X k =1 [Λ : ∆ k ] δ [Λ : Λ k ] ≤ n X k =1 O δk p (1+ δ ) ℓ k k Q kj =1 p ℓ j − ( j − n j ≤ n X k =1 p (1+ δ )( n + ℓ k ) k Q kj =1 p ℓ j − ( j − n j = n X k =1 p ( k + δ ) n +( δ ) ℓ k k Q k − j =1 p ℓ j − ( j − n j . We compute the ratio ( k -th term ) / (( k + 1) -th term ) of the series above: p ( k + δ ) n δ ) ℓkk Q k − j =1 p ℓj − ( j − n j p ( k + δ +1) n δ ) ℓk +1 k +1 Q kj =1 p ℓj − ( j − n j = p (1+ δ ) n +( δ +1) ℓ k k p ( k + δ +1) n + δℓ k +1 k +1 ≤ p (1+ δ ) n +( δ +1) ℓ k k p ( k + δ +1) n + δℓ k +1 k = p − kn + δ ( ℓ k − ℓ k +1 )+ ℓ k k Thus, if δ is sufficiently small and k is sufficiently large, we have that the exponent aboveis greater than 1 by (9). Hence, as p k is an increasing sequence of integers, the ratio testimplies that the resulting series above converges, and R ˆ M ( g ) dµ is finite. We conclude thatthe normal residual average of Λ is finite, as desired. (cid:3) Expected depth of random walks on groups
Fix for the whole section a finitely generated residually finite group G and a finite sym-metric generating set S . Consider the simple lazy random walk ( X n ) n ≥ on the Cayley graph XPECTED DEPTH OF RANDOM WALKS ON GROUPS 9
Cay(
G, S ) , as defined in §2.2. Recall that we are interested in E [ D G ( X n )] = X k ≥ k P [ D ( X n ) = k ] = X k ≥ P [ D ( X n ) > k ] = X k ≥ P ( X n ∈ Λ k \ { e } ) . The second equality is obtained through the same double-sum argument as in the proof ofLemma 3.1.4.1.
First estimates.Proposition 4.1.
We have lim inf n →∞ E [ D ( X n )] ≥ X k ≥ | G : Λ k | . Proof.
Recall the expression (1) for E [ D ( X n )] : E [ D ( X n )] = X k ≥ P ( X n ∈ Λ k \ { e } ) . Also recall from Corollary 2.5 that P ( X n ∈ Λ k \ { e } ) −−−→ n →∞ ( G :Λ k ] if k ≥ , if k = 0 , .The result follows from Fatou’s lemma. (cid:3) Corollary 4.2.
Suppose G is such that P k ≥ | G :Λ k | diverges. Then lim n →∞ E ( D ( X n )) = ∞ .Proof. This is a direct consequence of Proposition 4.1. (cid:3)
Proposition 4.3.
Suppose there exists a sequence of positive numbers { p k } k ≥ with • P k ≥ p k < ∞ ; • P ( X n ∈ Λ k \ { e } ) ≤ p k for all n ≥ and k ≥ .Then lim n →∞ E [ D ( X n )] = 2 + X k ≥ | G : Λ k | < ∞ . Proof.
Fix a sequence ( p k ) k ≥ as above, and set p = p = 1 . Then the convergence of P ( X n ∈ Λ k \ { e } ) to G :Λ k ] is dominated by p k . Since p k is summable, the dominatedconvergence theorem implies the desired result. (cid:3) Below, when applying Proposition 4.3, we will do so using the sequence p k = sup n ≥ P ( X n ∈ Λ k \ { e } ) , for k ≥ . This sequence obviously satisfies the domination criterion; one needs to show it is summablein order to apply the proposition.
XPECTED DEPTH OF RANDOM WALKS ON GROUPS 10
Sufficient condition using spectral properties.
Proof of Theorem 1.1.
Fix a linear group G with Property ( T ) . We will apply Proposition 4.3to show the desired convergence. Fix some k ≥ and let us bound P ( X n ∈ Λ k \ { e } ) forarbitrary n .First notice that, X n ∈ B SG ( n ) and therefore D G ( X n ) ≤ F SG ( n ) . It follows that P ( X n ∈ Λ k \ { e } ) = 0 if F SG ( n ) ≤ k. Suppose now that n is such that F SG ( n ) > k . Recall from Corollary 2.5 that | P ( X n ∈ Λ k ) − | G :Λ k | | ≤ µ nk , where µ k is the second largest eigenvalue of the transition matrix of the inducedlazy random walk on Cay( G/ Λ k , S ) . Now, since G has Property ( T ) , there exists a constant < θ < such that, for any normal finite index subgroup N ⊳ G , the second largest eigenvalueof the Cayley graph of G/N is bounded above by θ < (see [BdlHV08]; the exact value of θ does depend on the generating set S of G ). In particular, P ( X n ∈ Λ k \ { e } ) ≤ P ( X n ∈ Λ k ) ≤ | G : Λ k | + µ nk ≤ | G : Λ k | + θ n . Observe that the right-hand side above is decreasing in n , and therefore is maximal when n is minimal. Set N k = inf { n ≥ F SG ( n ) > k } . Then, by the above two cases, we deduce that P ( X n ∈ Λ k \ { e } ) ≤ | G : Λ k | + θ N k =: p k for all n ≥ and k ≥ . The values ( p k ) k ≥ defined above satisfy the second property of Proposition 4.3; we will shownow that they also satisfy the first.By [BRM15], there exist b ∈ N and C > such that F SG ( n ) ≤ Cn b for all n ≥ . Inparticular, for any k ≥ , N k ≥ C ′ k /b for some constant C ′ > that does not depend on k .Moreover, P k ≥ | G :Λ k | is finite by Lemma 3.1 and Theorem 3.2. Thus X k ≥ p k ≤ X k ≥ | G : Λ k | + X k ≥ θ C ′ k /b < ∞ . Applying Proposition 4.3 yields the desired result. (cid:3)
Sufficient condition: abelian groups.Lemma 4.4.
Let ( X n ) n ≥ be a lazy random walk on Z (that is on a Cayley graph of Z , as in§2.2). Then there exists a constant C > such that, for all m ≥ n ≥ P (cid:0) X n ∈ m Z \ { } (cid:1) ≤ C √ m . (11) In particular, there exists c > such that, for k ≥ , sup n ≥ P (cid:0) X n ∈ Λ k ( Z ) \ { } (cid:1) ≤ e − ck . (12) Proof.
We start with the proof of (11). Let c > be such that | X | < c almost surely.Below, write c m for the integer part of c m so as not to over-burden notation. Then, for n ≤ c m , P ( X n ∈ m Z \ { } ) = 0 . For n ≥ c m , write P ( X n ∈ m Z \ { } ) = X ℓ ∈ Z P (cid:0) X n ∈ m Z \ { } (cid:12)(cid:12) X n − c m = ℓ (cid:1) P ( X n − c m = ℓ ) . (13) XPECTED DEPTH OF RANDOM WALKS ON GROUPS 11
Now notice that, due to the choice of c , | X n − X n − c m | < m/ almost surely. However, forany fixed ℓ ∈ Z , there exists at most one element m ( ℓ ) ∈ m Z with | ℓ − m ( ℓ ) | < m/ . If nosuch element exists, choose m ( ℓ ) ∈ m Z arbitrarily. Thus P (cid:0) X n ∈ m Z \ { } (cid:12)(cid:12) X n − c m = ℓ (cid:1) ≤ P (cid:0) X n = m ( ℓ ) (cid:12)(cid:12) X n − c m = ℓ (cid:1) = P (cid:0) X c m = m ( ℓ ) − ℓ (cid:1) ≤ C √ c m , where the last inequality is due to [VSCC93, Thm. VI.5.1] and C > is some fixed constantdepending only on the transition probability of the random walk. When injecting the abovein (13), we find P ( X n ∈ m Z \ { } ) ≤ X ℓ ∈ Z C √ c m P ( X n − c m = ℓ ) = C √ c m . Since the right-hand side does not depend on n , this implies (11) with an adjusted value of C .We move on to proving (12). The (normal) subgroups of Z are of the form k Z , with k being their index. Thus, for k ≥ , Λ k ( Z ) = m k Z , where m k is the least common multiple of , . . . , k (see [BBRKM]). It follows from the PrimeNumber Theorem that there exists a constant c > such that m k ≥ exp( ck ) , ∀ k ≥ . The above bound, together with (11), implies (12) with an adjusted value of c . (cid:3) Corollary 4.5 (Expected depth for Z .) . Let ( X n ) n ≥ be a lazy random walk on Z (that is,on a Cayley graph of Z , as in §2.2). Then (14) E [ D Z ( X n )] −−−→ n →∞ X k ≥ | Z : Λ k | < ∞ . Proof.
For k ≥ , set p k = sup n ≥ P (cid:0) X n ∈ Λ k ( Z ) \ { } (cid:1) . By Lemma 4.4, P k ≥ p k < ∞ .Moreover, the sequence ( p k ) k ≥ dominates the convergence of P (cid:0) X n ∈ Λ k ( Z ) \ { } (cid:1) , asrequired in Proposition 4.3. The conclusion follows. (cid:3) Proposition 4.6.
Let G and H be two finitely generated residually finite groups. Let ( X n ) n ≥ be a random walk on a Cayley graph of G × H , as in §2.2. Then P [ D G × H ( X n ) > k ] ≤ P [ D G ( Y n ) > k ] + P [ D H ( Z n ) > k ] for all k ≥ ,where ( Y n ) n ≥ and ( Z n ) n ≥ are the random walks on G and H , respectively, induced by ( X n ) n ≥ .Proof. Notice that for g = ( g , g ) ∈ G × H we have estimates D G × H ( g ) ≤ D G ( g ) if g = eD G × H ( g ) ≤ D G ( g ) if g = e Therefore P [ D G × H ( X n ) > k ] ≤ P [ D G ( Y n ) > k ] + P [ D H ( Z n ) > k ] . (cid:3) XPECTED DEPTH OF RANDOM WALKS ON GROUPS 12
Corollary 4.7.
Let G be a finitely generated abelian group and let ( X n ) n ≥ be a lazy randomwalk on its Cayley graph, as in §2.2. Then E [ D G ( X n )] −−−→ n →∞ X k ≥ | G : Λ k | < ∞ , for any finite generating set S of G .Proof. Let G be a finitely generated abelian group. Then it may be written as G = Z × · · · × Z × H , where the product contains j copies of Z and H is a finite abelian group. The depthof elements of H is bounded by | H | . By Proposition 4.6 and Lemma 4.4, there exists c > such that P (cid:0) X n ∈ Λ k ( G ) \ { } (cid:1) ≤ je − ck , ∀ k > | H | . We conclude using the same domination argument as in the proof of Corollary 4.5. (cid:3)
Nilpotent groups: proof of Theorem 1.2.Torsion free case.
Suppose first that G is a finitely generated and torsion free nilpotentgroup, different from Z . Observe that G is a poly- C ∞ group, and in particular, G = Z ⋉ H where H is a non-trivial finitely generated nilpotent group. Let S be a system of generatorsof G and consider the lazy random walk on G with steps taken uniformly in S . We will writeit in the product form ( X n , Y n ) n ≥ , where X n ∈ Z and Y n ∈ H , for all n ≥ . Notice thenthat ( X n ) n ≥ is a lazy random walk on Z , as treated in Lemma 4.4. This is not true on thesecond coordinate: ( Y n ) n ≥ is not a random walk on H , it is not even a Markov process.For k ≥ , let p k ( G ) = sup n ≥ P (cid:2) D G ( X n , Y n ) > k (cid:3) , (15)and recall from Proposition 4.3 (and the commentary below it) that our goal is to prove that P k p k < ∞ .One may easily check that, for any m ∈ N , m Z ⋉ H is a normal subgroup of G . Thus, forall x ∈ Z \ { } and y ∈ H , D G ( x, y ) ≤ D Z ( x ) . We may therefore bound p k by p k ( G ) ≤ sup n ≥ P (cid:2) D G ( X n ) > k (cid:3) + P (cid:2) X n = 0 and D G (0 , Y n ) > k (cid:3) ≤ p k ( Z ) + sup n ≥ P (cid:2) X n = 0 and D G (0 , Y n ) > k (cid:3) . In the above, p k ( Z ) = sup n ≥ P (cid:2) D Z ( X n ) > k (cid:3) . We have shown in Lemma 4.4 that P k p k ( Z ) < ∞ , and we may focus on whether the second supremum is summable.Fix k ≥ . Since G is nilpotent, there exists c > such that F SG ( n ) ≤ c (log n ) h ( G ) , where h ( G ) is the Hirsch length of G (see Theorem 2.1). This should be understood as follows.In order for an element g ∈ G to have D G ( g ) ≥ k , it is necessary that k g k S ≥ exp( Ck /h ( G ) ) ,where k g k S denotes the length of g with respect to the generating set S and C > is aconstant independent of g . XPECTED DEPTH OF RANDOM WALKS ON GROUPS 13
In particular, we conclude that P (cid:2) X n = 0 and D G (0 , Y n ) > k (cid:3) = 0 if n < exp( Ck /h ( G ) ) ; P (cid:2) X n = 0 and D G (0 , Y n ) > k (cid:3) ≤ P ( X n = 0) if n ≥ exp( Ck /h ( G ) ) .In treating the second case, observe that, since ( X n ) n ≥ is a random walk on Z , P ( X n = 0) ≤ c n − / , for some constant c > (see [VSCC93, Thm. VI.5.1]). Thus P (cid:2) X n = 0 and D G (0 , Y n ) > k (cid:3) ≤ c exp (cid:16) − C k h ( G ) (cid:17) , ∀ n ∈ N . We conclude that X k ≥ sup n ≥ P (cid:2) X n = 0 and D G (0 , Y n ) > k (cid:3) < ∞ , and therefore that P k ≥ p k ( G ) < ∞ . General nilpotent case.
Let now G be a finitely generated nilpotent group. Consider theset T ( G ) of all torsion elements of G . Since G is nilpotent, the set T ( G ) is a finite normalsubgroup in G . Consider an epimorphism π : G → G/T ( G ) . Denote by H the quotient G/T ( G ) and notice that H is a finitely generated torsion-free nilpotent group.Observe that for any non-trivial element in H detected by a normal subgroup in H of index k there exists a normal subgroup in G of index at most k that detects its preimage. In otherwords, for all g ∈ G \ T ( G ) , D G ( g ) ≤ D H ( π ( g )) . The random walk ( X n ) n ≥ on G induces a random walk ( π ( X n )) n ≥ on H . Let d =max { D G ( g ) , g ∈ T ( G ) } . Due to the observation above, for all k > d , P [ D G ( X n ) ≥ k ] ≤ P [ D H ( π ( X n )) ≥ k ] . We deduce from the case of torsion free nilpotent groups that P k>d sup n ≥ P [ D G ( X n ) ≥ k ] < ∞ . Then the second point of Proposition 4.3 applies and we obtain the desired conclusion. (cid:3) A counter example.Proposition 4.8 (Groups with infinite expected depth) . There exists a finitely generatedresidually finite group G such that lim n →∞ E ( D ( X n )) = ∞ , but for which the “presumed” limit P k ≥ | G :Λ k | is finite.Proof. The existence of finitely generated residually finite groups with arbitrary large residualfiniteness growth was shown in [BRS16].Let H be a two-generated group (with generators a, b ) such that, for any n ≥ , there existsan element h n in the ball of radius n of H with D H ( h n ) ≥ n . Let ( X n ) n ≥ be a lazy simplerandom walk on the Cayley graph of G = H × Z with the natural choice of generators (thatis ( a, , ( b, and ( e H , , where a and b are the two generators of H mentioned above) andtheir inverses.Then, for any n ≥ , P [ X n = ( h n , ≥ − n . Therefore E [ D G ( X n )] ≥ P [ X n = ( h n , · D G [( h n , P [ X n = ( h n , · D H ( h n ) ≥ n , XPECTED DEPTH OF RANDOM WALKS ON GROUPS 14
Hence the expectation of the depth of X n tends to infinity.Furthermore, observe that Λ k ( G ) is a subgroup of H × Λ k ( Z ) and hence | G : Λ k ( G ) | ≥ | Z : Λ k ( Z ) | . It follows that P k ≥ | G :Λ k ( G ) | ≤ P k ≥ | Z :Λ k ( Z ) | < ∞ . (cid:3) References [BBRKM] I. Biringer, K. Bou-Rabee, M. Kassabov, and F. Matucci. Intersection growth in nilpotent groups.submitted, arXiv:math.GR/1309.7993.[BdlHV08] B. Bekka, P. de la Harpe, and A. Valette.
Kazhdan’s property (T) , volume 11 of
New MathematicalMonographs . Cambridge University Press, Cambridge, 2008.[BR10] K. Bou-Rabee. Quantifying residual finiteness.
J. Algebra , 323(3):729–737, 2010.[BRK12] K. Bou-Rabee and T. Kaletha. Quantifying residual finiteness of arithmetic groups.
Compos. Math. ,148(3):907–920, 2012.[BRM10] K. Bou-Rabee and D. B. McReynolds. Bertrand’s postulate and subgroup growth.
J. Algebra ,324(4):793–819, 2010.[BRM15] K. Bou-Rabee and D. B. McReynolds. Extremal behavior of divisibility functions.
Geom. Dedicata ,175:407–415, 2015.[BRS16] K. Bou-Rabee and B. Seward. Arbitrarily large residual finiteness growth.
J. Reine Angew. Math. ,710:199–204, 2016.[BV02] J. Burillo and E. Ventura. Counting primitive elements in free groups.
Geom. Dedicata , 93:143–162,2002.[Hal69] P. Hall.
The Edmonton notes on nilpotent groups . Queen Mary College Mathematics Notes. Math-ematics Department, Queen Mary College, London, 1969.[Pan08] P. Pansu. Croissance des boules et des géodésiques fermées dans les nilvariétés.
Ergodic Theoryand Dynamical Systems , 3(3):415–445, Sep 2008.[Seg83] D. Segal.
Polycyclic groups , volume 82 of
Cambridge Tracts in Mathematics . Cambridge UniversityPress, Cambridge, 1983.[VSCC93] N. T. Varopoulos, L. Saloff-Coste, and T. Coulhon.
Analysis and Geometry on Groups: . CambridgeTracts in Mathematics. Cambridge University Press, Jan 1993.[Wil98] J. S. Wilson.
Profinite groups , volume 19 of
London Mathematical Society Monographs. New Series .The Clarendon Press, Oxford University Press, New York, 1998.
School of Mathematics, CCNY CUNY, New York City, New York, USA
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