Geometry of the word problem for 3-manifold groups

TL;DR

This paper introduces an algorithmic framework for solving the word problem in all closed 3-manifold groups by demonstrating they are autostackable, and extends these ideas to related group structures and geometries.

Contribution

It establishes that fundamental groups of closed 3-manifolds are autostackable and develops new concepts like autostackability respecting subgroups, broadening the scope of algorithmic solutions.

Findings

All fundamental groups of closed 3-manifolds are autostackable.

Groups hyperbolic relative to abelian subgroups are strongly coset automatic.

Fundamental groups of compact geometric 3-manifolds with torus boundary are autostackable respecting peripheral subgroups.

Abstract

We provide an algorithm to solve the word problem in all fundamental groups of closed 3-manifolds; in particular, we show that these groups are autostackable. This provides a common framework for a solution to the word problem in any closed 3-manifold group 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.