Subgroup distortion in wreath products of cyclic groups

Abstract

We 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 .