Subgroup distortion in wreath products of cyclic groups

AbstractWe study the effects of subgroup distortion in the wreath products AwrZ, where A is finitely generated abelian. We show that every finitely generated subgroup of AwrZ has distortion function equivalent to some polynomial. Moreover, for A infinite, and for any polynomial lk, there is a 2-generated subgroup of AwrZ having distortion function equivalent to the given polynomial. Also, a formula for the length of elements in arbitrary wreath product HwrG easily shows that the group Z2wrZ2 has distorted subgroups, while the lamplighter group Z2wrZ 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