# Kristen Pueschel

### First Position

Postdoctoral Associate, University of Arkansas### Dissertation

*On Residual Properties of Groups and Dehn Functions for Mapping Tori of Right Angled Artin Groups*

### Advisor

### Research Area

### Abstract

Given a group G and an automorphism φ : G → G, the algebraic mapping torus Mφ, is the HNN-extension of G where the stable letter conjugates g to φ(g). In this thesis, we study mapping tori with base groups G that are right angled Artin groups on few generators. In particular, we examine how the Dehn function of a mapping torus depends on the automorphism φ used to generate the group when G is F2 × Z or Z 2 ∗ Z. We then extend our methods to produce bounds for the Dehn functions of other mapping tori with few-generator base groups. A group G is residually finite if for every g ∈ G − {1}, there is a finite quotient of G in which the image of g is non-trivial. The hydra are a family of groups with fast-growing Dehn functions; the Dehn function of each group is equivalent to an Ackermann function. In this thesis we show that hydra are not residually finite, answering a question of Kharlampovich, Myasnikov, and Sapir. A group G is residually solvable if for every g ∈ G−{1}, there is a solvable quotient of G in which the image of g is non-trivial. In this thesis we introduce and explore properties of the function that measures the smallest derived length of a solvable quotient in which g survives, in terms of the length of g.