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Title page for ETD etd-03222011-131219

Type of Document Dissertation
Author Davis, Tara Colleen
URN etd-03222011-131219
Title Subgroup Distortion in Metabelian and Free Nilpotent Groups
Degree PhD
Department Mathematics
Advisory Committee
Advisor Name Title
Alexander Olshanskii Committee Chair
Denis Osin Committee Member
Guoliang Yu Committee Member
Mark Sapir Committee Member
Senta Victoria Greene Committee Member
  • membership problem
  • wreath product
  • subgroup distortion
Date of Defense 2011-03-21
Availability unrestricted
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).

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