Subgroup distortion in wreath products of cyclic groups

Abstract

a b s t r a c tWe study the effects of subgroup distortion in the wreath products A wr Z, where A is finitely generated abelian. We show that every finitely generated subgroup of A wr Z has distortion function equivalent to some polynomial. Moreover, for A infinite, and for any polynomial l k , there is a 2-generated subgroup of A wr Z having distortion function equivalent to the given polynomial. Also, a formula for the length of elements in arbitrary wreath product H wr G easily shows that the group Z 2 wr Z 2 has distorted subgroups, while the lamplighter group Z 2 wr Z has no distorted (finitely generated) subgroups. In the course of the proof, we introduce a notion of distortion for polynomials. We are able to compute the distortion of any polynomial in one variable over Z, R or C.

On conjugacy separability of graph products of groups

“…Following Lemmas 5.8 and 5.9, we see that there is an element t ∈ B such thatg(b, t) = a = 1. Since g ≤ n, we have by [5,Theorem 3.4] that | supp(g)| ≤ n. As B is residually-C and b t is C-separable, we have by Lemma 5.10 that there is a subgroup N B ∈ N C (B) such that π NB is injective on supp(g), π NB ( b t 1 ) = π NB ( b t 2 ) for t 1 , t 2 ∈ supp(g), and |B/N B | ≤ (Cyclic B,C (n)) n 2 . Letting π : A ≀ B → A ≀ (B/N B ) be the natural extension given by Lemma 4.3, we have that π(f )(b, t) = a. Lemma 5.9 implies that…”

“…By assumption, if c ∈ C, then c / ∈ C B (b). We have by [5,Theorem 3.4] that if a ∈ supp(f ) ∪ supp(g), then there exists a constant C 5 > 0 such that a S ≤ C 5 n.…”

Rational subsets and submonoids of wreath products

“…Setting a ′ = 0 and b ′ = b shows that (13) has the form required in the conclusion of Claim 5. Next, assume that y j+1 has the form (5). Hence (p, +1, q) ∈ δ and I(m, n)Y j+1 is of the form [0, 0, #, 0, #, 0, #, .…”

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

“…To do this we take advantage of the structure of polycyclic groups in conjunction with the idea of subgroup distortion. Subgroup distortion has been much studied in geometric group theory [Ger96,DO10]; for polycyclic groups, the primary reference is Osin [Osi02].…”