The development of the theory of automatic groups

TL;DR

This paper reviews the development of automatic group theory, covering definitions, properties, key classes like hyperbolic groups, and computational tools, highlighting advances and open problems in the field.

Contribution

It provides a comprehensive overview of automatic groups, including new results on hyperbolic and 3-manifold groups, and discusses software and algorithms for computing automatic structures.

Findings

Hyperbolic groups are proven to be automatic.

Software developed at Warwick computes automatic structures.

Automaticity is established for various infinite group families.

Abstract

We describe the development of the theory of automatic groups. We begin with a historical introduction, define the concepts of automatic, biautomatic and combable groups, derive basic properties, then explain how hyperbolic groups and the groups of compact 3-manifolds based on six of Thurston's eight geometries can be proved automatic. We describe software developed in Warwick to compute automatic structures, as well as the development of practical algorithms that use those structures. We explain how actions of groups on spaces displaying various notions of negative curvature can be used to prove automaticity or biautomaticity, and show how these results have been used to derive these properties for groups in some infinite families (braid groups, mapping class groups, families of Artin groups, and Coxeter groups). Throughout the text we flag up open problems as well as problems that remained open for some time but have now been resolved.