Subgroup Distortion in Wreath Products of Cyclic Groups

TL;DR

This paper investigates subgroup distortion in wreath products of cyclic groups, revealing that finitely generated subgroups exhibit polynomial distortion, with specific examples illustrating the presence or absence of distorted subgroups.

Contribution

It characterizes the distortion functions of finitely generated subgroups in wreath products of abelian groups and provides explicit examples of distorted and undistorted subgroups.

Findings

Finitely generated subgroups have polynomial distortion functions.

Existence of subgroups with any polynomial distortion in certain wreath products.

Some wreath products, like Z_2 wr Z, have no distorted finitely generated subgroups.

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.