Groups defined by automata

TL;DR

This paper reviews how finite automata are used to define and analyze infinite groups, highlighting two main approaches: automatic groups and automata groups, each revealing unique geometric and algebraic properties.

Contribution

It provides a comprehensive overview of automata-based methods for constructing and studying infinite groups, emphasizing recent developments and their implications.

Findings

Automata define normal forms and group operations effectively.

Automata groups exhibit exotic behaviors, expanding understanding of infinite groups.

Connections between automata, geometry, and group theory are deepened.

Abstract

This is Chapter 24 in the "AutoMathA" handbook. Finite automata have been used effectively in recent years to define infinite groups. The two main lines of research have as their most representative objects the class of automatic groups (including word-hyperbolic groups as a particular case) and automata groups (singled out among the more general self-similar groups). The first approach implements in the language of automata some tight constraints on the geometry of the group's Cayley graph, building strange, beautiful bridges between far-off domains. Automata are used to define a normal form for group elements, and to monitor the fundamental group operations. The second approach features groups acting in a finitely constrained manner on a regular rooted tree. Automata define sequential permutations of the tree, and represent the group elements themselves. The choice of particular classes of automata has often provided groups with exotic behaviour which have revolutioned our perception of infinite finitely generated groups.