Languages, Geodesics, and HNN Extensions

Authors

• Maranda Franke, University of Nebraska-LincolnFollow

First Advisor

Mark Brittenham

Second Advisor

Susan Hermiller

Date of this Version

5-2017

Document Type

Dissertation

Citation

Maranda Franke. Languages, geodesics, and HNN extensions. Ph.D. dissertation, University of Nebraska - Lincoln, 2017.

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 Brittenahm and Susan Hermiller. Lincoln, Nebraska: May, 2017

Copyright (c) Maranda Franke

Abstract

The complexity of a geodesic language has connections to algebraic properties of the group. Gilman, Hermiller, Holt, and Rees show that a finitely generated group is virtually free if and only if its geodesic language is locally excluding for some finite inverse-closed generating set. The existence of such a correspondence and the result of Hermiller, Holt, and Rees that finitely generated abelian groups have piecewise excluding geodesic language for all finite inverse-closed generating sets motivated our work. We show that a finitely generated group with piecewise excluding geodesic language need not be abelian and give a class of infinite non-abelian groups which have piecewise excluding geodesic languages for certain generating sets. The quaternion group is shown to be the only non-abelian 2-generator group with piecewise excluding geodesic language for all finite inverse-closed generating sets. We also show that there are virtually abelian groups with geodesic languages which are not piecewise excluding for any finite inverse-closed generating set.

Autostackable groups were introduced by Brittenham, Hermiller, and Holt as a generalization of asynchronously automatic groups on prefix-closed normal forms and groups with finite convergent rewriting systems. Brittenham, Hermiller, and Johnson show that Stallings' non-FP_3 group, an HNN extension of a right-angled Artin group, is autostackable. We extend this autostackability result to a larger class of HNN extensions of right-angled Artin groups.

Advisers: Mark Brittenahm and Susan Hermiller