Subgroup Distortion in Wreath Products of Cyclic Groups

Details Subgroup Distortion in Wreath Products of Cyclic Groups Subgroup Distortion in Wreath Products of Cyclic Groups

2010-09-22

arxiv.org/abs/1009.4364v3

Mathematics

Group Theory

Scientific paper

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.

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