Mathematics, Department of
First Advisor
Susan Hermiller
Date of this Version
5-2023
Citation
Ash DeClerk. Prefix-rewriting: the falsification by fellow traveler property and practical computations. Ph.D. dissertation, University of Nebraska - Lincoln, 2023.
Abstract
The word problem is one of the fundamental areas of research in infinite group theory, and rewriting systems (including finite convergent rewriting systems, automatic structures, and autostackable structures) are key approaches to working on the word problem. In this dissertation, we discuss two approaches to creating bounded regular convergent prefix-rewriting systems.
Groups with the falsification by fellow traveler property are known to have solvable word problem, but they are not known to be automatic or to have finite convergent rewriting systems. We show that groups with this geometric property are geodesically autostackable. As a key part of proving this, we show that a wider class of groups, namely groups with a weight non-increasing synchronously regular convergent prefix-rewriting system, have a bounded regular convergent prefix-rewriting system.
Our second approach to creating prefix-rewriting systems is a more general approach. We design a procedure that, when provided with a finitely presented group G = < A | R > and an ordering < on A*, searches for a bounded convergent prefix-rewriting system. We also create a class of orderings for which each step of this procedure can be practically computed, and which guarantees that any bounded convergent prefix-rewriting system is an autostackable structure.
Adviser: Susan Hermiller
Comments
A DISSERTATION Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Fulfilment of Requirements For the Degree of Doctor of Philosophy, Major: Mathematics, Under the Supervision of Professor Susan Hermiller. Lincoln, Nebraska: May, 2023
Copyright © 2023 Ash DeClerk