From automatic structures to automatic groups

TL;DR

This paper introduces Cayley graph automatic groups, a broader class than automatic groups, with favorable properties like invariance under generators and quadratic-time Word Problem decidability, expanding the scope of automatic group theory.

Contribution

The paper generalizes automatic groups to graph automatic groups, demonstrating their properties and including new classes such as finitely generated 2-nilpotent and Baumslag-Solitar groups.

Findings

Graph automatic groups are invariant under generator change.

They are closed under certain group operations.

Word Problem is decidable in quadratic time.

Abstract

In this paper we introduce the concept of a Cayley graph automatic group (CGA group or graph automatic group, for short) which generalizes the standard notion of an automatic group. Like the usual automatic groups graph automatic ones enjoy many nice properties: these group are invariant under the change of generators, they are closed under direct and free products, certain types of amalgamated products, and finite extensions. Furthermore, the Word Problem in graph automatic groups is decidable in quadratic time. However, the class of graph automatic groups is much wider then the class of automatic groups. For example, we prove that all finitely generated 2-nilpotent groups and Baumslag-Solitar groups B(1,n) are graph automatic, as well as many other metabelian groups.