Distortion of wreath products in some finitely presented groups

Wreath products such as Z o Z are not finitely presentable yet can occur as subgroups of finitely presented groups. Here we compute the distortion of ZoZ as a subgroup of Thompson’s group F and as a subgroup of Baumslag’s metabelian group G. We find that Z o Z is undistorted in F but is at least exponentially distorted in G. 1