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Mathematische Zeitschrift Math. Z. (2015) 279:811–848 DOI 10.1007/s00209-014-1395-2 Mathematische Zeitschrift Random walks on free solvable groups Laurent Saloff-Coste · Tianyi Zheng Received: 30 October 2013 / Accepted: 5 August 2014 / Published online: 4 November 2014 © Springer-Verlag Berlin Heidelberg 2014 Abstract For any finitely generated group G,letn → G (n) be the function that describes the rough asymptotic behavior of the probability of return to the identity element at time 2n of a symmetric simple random walk on G (this is an invariant of quasi-isometry). We determine this function when G is the free solvable group Sd,r of derived length d on r generators and some related groups. Mathematics Subject Classification 20F69 · 60J10 1 Introduction 1.1 The random walk group invariant G Let G be a finitely generated group. Given a probability measure μ on G, the random walk driven by μ (started at the identity element e of G)istheG-valued random process = ξ ...ξ (ξ )∞ Xn 1 n where i 1 is a sequence of independent identically distributed G-valued μ ∗ v( ) = ( )v( −1 ) random variables with law .Ifu g h u h h g denotes the convolution of two μ (n) functions u and v on G then the probability that Xn = g is given by Pe (Xn = g) = μ (g) where μ(n) is the n-fold convolution of μ. Given a symmetric set of generators S, the word-length |g| of g ∈ G is the minimal length of a word representing g in the elements of S. The associated volume growth function, r → VG,S(r), counts the number of elements of G with |g|≤r. The word-length induces a left invariant metric on G which is also the graph metric on the Cayley graph (G, S).A Laurent Saloff-Coste and Tianyi Zheng authors partially supported by NSF Grant DMS 1004771. L. Saloff-Coste (B) Department of Mathematics, Cornell University, Ithaca, NY, USA e-mail: [email protected] T. Zheng Department of Mathematics, Stanford University, Stanford, CA, USA 123 812 L. Saloff-Coste, T. Zheng quasi-isometry between two Cayley graphs (Gi , Si ), i = 1, 2, say, from G1 to G2,isamap q : G1 → G2 such that −1 C d2(q(x), q(y)) ≤ d1(x, y) ≤ C(1 + d2(q(x), q(y))) { ( , ( )) ≤ and supg,∈G2 d2 g q G1 C for some finite positive constant C. This induces an equiv- alence relation on Cayley graphs. In particular, (G, S1), (G, S2) are quasi-isometric for any choice of generating sets S1, S2. See, e.g., [5] for more details. Given two monotone functions φ,ψ, write φ ψ if there are constants ci ∈ (0, ∞), 1 ≤ i ≤ 4, such that c1ψ(c2t) ≤ φ(t) ≤ c3ψ(c4t) (using integer values if φ,ψ are defined on N). We denote by and the associated inequalities. If S1, S2 are two symmetric generating sets for G,thenVG,S1 VG,S2 .WeusethenotationVG to denote either the -equivalence class of VG,S or any one of its representatives. The volume growth function VG is one of the simplest quasi-isometry invariant of a group G. −1 By [16, Theorem 1.4], if μi , i = 1, 2, are symmetric (i.e., μi (g) = μi (g ) for all g ∈ G) finitely supported probability measures with generating support, then the functions → φ ( ) = μ(2n)( ) φ φ n i n i e satisfy 1 2. By definition, we denote by G any function that belongs to the -equivalence class of φ1 φ2. In fact, G is an invariant of quasi-isometry. Further, ifμ is a symmetric probability measure with generating support and finite second | |2μ( )<∞ μ(2n)( ) ( ) moment G g g then e G n .See[16]. 1.2 Free solvable groups This work is concerned with finitely generated solvable groups. Recall that G(i), the derived series of G, is defined inductively by G(0) = G, G(i) =[G(i−1), G(i−1)]. A group is solvable if G(i) ={e} for some i and the smallest such i is the derived length of G. A group G is polycyclic if it admits a normal descending series G = N0 ⊃ N1 ⊃ ··· ⊃ Nk ={e} such that each of the quotient Ni /Ni+1 is cyclic. The lower central series γ j (G), j ≥ 1, of a group G is obtained by setting γ1(G) = G and γ j+1 =[G,γj (G)]. A group G is nilpotent of nilpotent class c if γc(G) ={e} and γc+1(G) ={e}. Finitely generated nilpotent groups are polycyclic and polycyclic groups are solvable. Recall the following well-known facts. If G is a finitely generated solvable group then D either G has polynomial volume growth VG (n) n for some D = 0, 1, 2,...,orG has exponential volume growth VG (n) exp(n). See, e.g., [5] and the references therein. If D −D/2 VG (n) n then G is virtually nilpotent and G (n) n .IfG is polycyclic with 1/3 exponential volume growth then G (n) exp(−n ).See[1,12,22–24] and the references given there. However, among solvable groups of exponential volume growth, many other behaviors than those described above are known to occur. See, e.g., [8,15,20]. Our main result is the following theorem. Set = ( + ) ( ) = ( + ). log[1] n log 1 n and log[i] n log 1 log[i−1] n Theorem 1.1 Let Sd,r be the free solvable group of derived length d on r generators, that (d) is, Sd,r = Fr /Fr where Fr is the free group on r generators, r ≥ 2. • If d = 2 (the free metabelian case) then ( ) − r/(r+2)( )2/(r+2) . S2,r n exp n log n 123 Random walks on free solvable groups 813 • If d > 2 then ⎛ ⎞ 2/r log n ( ) ⎝− [d−1] ⎠ . Sd,r n exp n log[d−2] n In the case d = 2, this result is known and due to Anna Erschler who computed the Følner function of S2,r in an unpublished work based on the ideas developed in [8]. We give a different proof. Note that if G isr-generated and solvable of length at most d then there exists c, k ∈ (0, ∞) ( ) ≥ ( ) such that G n c Sd,r kn . 1.3 On the groups of the form Fr /[N, N] The first statement in Theorem 1.1 can be generalized as follows. Let N be a normal subgroup (d−1) of Fr and consider the tower of r generated groups d (N) defined by d (N) = Fr /N . Given information about 1(N) = Fr /N, more precisely, about the pair (Fr , N), one may =[ , ] ( ) = Zr hope to determine d (N) (in Theorem 1.1, N Fr Fr and 1 N ). Here, it is important to note that the groups d (N), d ≥ 2, depend not only of the group 1(N) but also of the particular presentation Fr /N = 1(N) of that group that is chosen. See Example 2.3. The following theorem captures some of the results we obtain in this direction when d = 2. Further examples are given in Sect. 5.3. Theorem 1.2 Let N Fr , 1(N) = Fr /N and 2(N) = Fr /[N, N] as above. • Assume that r ≥ 2 and that 1(N) be infinite nilpotent of volume growth of degree D. Then we have ( ) − D/(D+2)( )2/(D+2) . 2(N) n exp n log n • Assume that q −n n – either 1(N) = Zq Z with presentation a, t|a , [a, t at ], n ∈ Z, −1 q – or 1(N) = BS(1, q) with presentation a, b|a ba = b . Then we have n ( )(n) exp − . 2 N (log n)2 In Sect. 5, Theorem 5.4, we treat polycyclic groups of exponential volume growth equipped with their standard polycyclic presentation. Obtaining results for d ≥ 3 is not easy. We treat a few examples beyond the case N = [Fr , Fr ] of Theorem 1.1. These examples include the case when N = γc(Fr ). See Theorem 6.13. Remark 1.3 Fix the presentation Fr /N = 1(N).Letμ be the probability measure driving (ξ )∞ the lazy simple random walk n 0 on Fr so that μ(ξ = ) = μ(n)( ). Pe n g g = ( )∞ = ( )∞ ( ) ( ) Let X Xn 0 and Y Yn 0 be the projections on 2 N and 1 N , respectively so that ( ) μ( = ) ( ) μ( =¯) 2(N) n Pe Xn e and 1(N) n Pe Yn e 123 814 L. Saloff-Coste, T. Zheng where e (resp. e¯) is the identity element in 2(N) (resp. 1(N).) By the flow interpretation of the group 2(N) developed in [6,14] and reviewed in Sect. 2.2 below, μ( = ) = μ( ∈ B ) Pe Xn e Pe Y n where Bn is the event that, at time n, every oriented edge of the marked Cayley graph 1(N) r has been traversed an equal number of times in both directions. For instance, if 1(N) = Z , ( ) (− r/(2+r)( )2/(2+r)) the estimate 2(N) n exp n log n also gives the order of magnitude of the probability that a simple random walk on Zr returns to its starting point at time n having crossed each edge an equal number of time in both direction. 1.4 Other random walk invariants Let |g| be the word-length of G with respect to some fixed finite symmetric generating set α and ρα(g) = (1 +|g|) .In[3], for any finitely generated group G and real α ∈ (0, 2),the non-increasing function G,ρα : N n → G,ρα (n) ∈ (0, ∞) is defined in such a way that it provides the best possible lower bound (2n) ∃ c, k ∈ (0, ∞), ∀ n,μ (e) ≥ cG,ρα (kn), valid for every symmetric probability measure μ on G satisfying the weak-ρα-moment con- dition W(ρα,μ)= sup{sμ({g : ρα(g)>s})} < ∞.
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