Closure and homological properties of (auto)stackable groups

Authors

Ashley Johnson, University of Nebraska - LincolnFollow

Date of this Version

8-2013

Comments

A DISSERTATION Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Fulfillment of Requirements For the Degree of Doctor of Philosophy, Major: Mathematics, Under the Supervision of Professors Mark Brittenham and Susan Hermiller. Lincoln, Nebraska: August, 2013

Copyright (c) 2013 Ashley Johnson

Abstract

Let G be a finitely presented group with Cayley graph Γ. Roughly, G is a stackable group if there is a maximal tree T in Γ and a function φ, defined on the edges in Γ, for which there is a natural ‘flow’ on the edges in Γ\T towards the identity. Additionally, if graph (φ), which consists of pairs (e; φ(e)) for e an edge in Γ, forms a regular language, then G is autostackable. In 2011, Brittenham and Hermiller introduced stackable groups in [4], in part, as a means to gain traction on the word problem for 3-manifold groups. They showed that if graph(φ) is (at least) decidable, as a language, then there is an effective algorithm which solves the word problem; furthermore, they show that stackable groups have an inductive procedure for building van Kampen diagrams, which helps provide insight into the complexity of the word problem.

As one part of my thesis research, I consider group constructions under which the (auto)stackable property is preserved. In this thesis, I show positive results in the case of graph products (a generalization of direct and free products), group extensions and finite index supergroups, and in the case of free products with amalgamation of free abelian groups over an infinite cyclic group. Using closure under group extensions, I also show that polycyclic groups are autostackable, and that there exists an autostackable group with unsolvable conjugacy problem.

Autostackable groups generalize the structures of automatic groups and groups with finite complete rewriting systems, both of which are known to be of type FP_∞. However, in this paper, I show that there exists an autostackable group that is not of type FP₃.

Advisers: Mark Brittenham and Susan Hermiller