GEOMETRY OF THE WORD PROBLEM FOR 3-MANIFOLD GROUPS

Authors

• Mark Brittenham, University of Nebraska - LincolnFollow
• Susan Hermiller, University of Nebraska - LincolnFollow
• Tim Susse, Bard CollegeFollow

Document Type

Article

Date of this Version

2017

Citation

Volume 499, 1 April 2018, Pages 111-150

Comments

Published in the Journal of Algebra

https://doi.org/10.1016/j.jalgebra.2017.12.001

Abstract

We provide an algorithm to solve the word problem in all fundamental groups of 3-manifolds that are either closed, or compact with (finitely many) boundary compo- nents consisting of incompressible tori, by showing that these groups are autostackable. In particular, this gives a common framework to solve the word problem in these 3-manifold groups using finite state automata. We also introduce the notion of a group which is autostackable respecting a subgroup, and show that a fundamental group of a graph of groups whose vertex groups are autostackable respecting any edge group is autostackable. A group that is strongly coset automatic over an autostackable subgroup, using a prefix-closed transversal, is also shown to be autostackable respecting that subgroup. Building on work by Antolin and Ciobanu, we show that a finitely generated group that is hyperbolic relative to a collection of abelian subgroups is also strongly coset automatic relative to each subgroup in the collection. Finally, we show that fundamental groups of compact geometric 3-manifolds, with boundary consisting of (finitely many) incompressible torus components, are autostackable respecting any choice of peripheral subgroup.