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 AbstractThe 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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