Algorithms and topology for Cayley graphs of groups

TL;DR

This paper introduces autostackability, a topological property of Cayley graphs for finitely generated groups, which ensures solvable word problems and effective diagram construction, connecting it with automatic groups and rewriting systems.

Contribution

It provides a new topological framework for understanding group properties via Cayley graphs, linking autostackability with existing concepts like automatic groups and rewriting systems.

Findings

Autostackability implies solvable word problem.

Groups with finite complete rewriting systems are autostackable.

Fundamental groups of all closed 3-manifolds are autostackable.

Abstract

Autostackability for finitely generated groups is defined via a topological property of the associated Cayley graph which can be encoded in a finite state automaton. Autostackable groups have solvable word problem and an effective inductive procedure for constructing van Kampen diagrams with respect to a canonical finite presentation. A comparison with automatic groups is given. Another characterization of autostackability is given in terms of prefix-rewriting systems. Every group which admits a finite complete rewriting system or an asynchronously automatic structure with respect to a prefix-closed set of normal forms is also autostackable. As a consequence, the fundamental group of every closed 3-manifold with any of the eight possible uniform geometries is autostackable.