The rate of escape for random walks on polycyclic and metabelian groups

Abstract

We use subgroup distortion to determine the rate of escape of a simple random walk on a class of polycyclic groups, and we show that the rate of escape is invariant under changes of generating set for these groups. For metabelian groups, we define a stronger form of subgroup distortion, which applies to non-finitely generated subgroups. Under this hypothesis, we compute the rate of escape for certain random walks on metabelian groups via a comparison to the toppling of a dissipative abelian sandpile.

Speed Exponents of Random Walks on Groups

“…(See also Thompson (2010) for more classes with this behavior.) On nonamenable groups, on the other hand, we have…”

A Law of Iterated Logarithm on Lamplighter Diagonal Products

“…In [18], Revelle studies groups of the form G ≀ Z, where the random walk on G has α-tight degree of escape, and proves that G ≀ Z has α ′ = 1+α 2 -tight degree of escape; he also shows that n α ′ (log log n) 1−α ′ is an upper scaling function for G ≀ Z, whereas n α ′ / (log log n) 1−α ′ is a lower scaling function for this group. Revelle studies in addition several Baumstag-Soliter groups, proving they have an inner radius of order n/ log log n and an outer radius of order √ n log log n. Finally, in [20] Thompson proves Laws of Iterated Logarithm for certain polycyclic and metaabelian groups. He shows that these groups are all diffusive, have 1 2 -tight degree of escape (which is a form of control over the tail of the distance function), and have an upper scaling function g(n) = √ n log log n.…”

Finite-dimensional representations constructed from random walks

“…It suffices to show that there is a constant D ≥ 1 with the following property (cf. [35,Lemma 3.4]). Let s i ∈ G be such that |s i | G ≤ 1 and put g k := s 1 · · · s k ∈ G and M n := max k=1,...,n |g k N | G/N .…”