Subgroup Distortion in Metabelian and Free Nilpotent Groups

Abstract

The main result of this dissertation sheds light on subgroup distortion in metabelian and free nilpotent groups.

A subgroup of a finitely generated free nilpotent group F is undistorted if and only if it is a retract of a subgroup of finite index in F. Also, the effects of subgroup distortion in the wreath products A wr Z, where

A is finitely generated abelian are studied. It is shown 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 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.

Following the work done by Olshanskii for groups, it is also described, for a given semigroup S, which functions l : S → N can be realized up to equivalence as length functions g ↦→ |g|H by embedding S into a finitely generated semigroup H. Following the work done by Olshanskii and Sapir, a complete description of length functions of a given finitely generated semigroup with enumerable set of relations inside a finitely presented semigroup is provided.

This classification for groups has connections with another function of interest in geometric group theory: the relative growth function. There are connections between the relative growth of cyclic subgroups, and the corresponding distortion function of the embedding. In particular, when the distortion is non-recursive, the relative growth is at least almost quadratic. On the other hand, there exists a cyclic subgroup of a two generated group such that the distortion function associated to the embedding is not bounded above by any recursive function, and yet the relative growth is o(r^2).