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 lk, 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 Z2 wr Z2 has distorted subgroups, while the lamplighter group Z2 wr Z has no distorted (finitely generated) subgroups.
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