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
Keywords	membership problem wreath product subgroup distortion
Date of Defense	2011-03-21
Availability	unrestricted
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).
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