# Quasi-automatic groups are asynchronously automatic

## TL;DR

This paper proves that quasi-automatic groups are also asynchronously automatic, establishing an equivalence between these two classes of groups under certain rationality conditions.

## Contribution

It demonstrates that quasi-automatic groups are asynchronously automatic, clarifying the relationship between these two notions in group theory.

## Key findings

- Quasi-automatic groups are asynchronously automatic.
- The equivalence holds specifically for groups, not just semigroups.
- Provides a characterization of automaticity in certain algebraic structures.

## Abstract

A quasi-automatic semigroup is a finitely generated semigroup with a rational set of representatives such that the graph of right multiplication by any generator is a rational relation. A asynchronously automatic semigroup is a quasi-automatic semigroup for which these rational relations are also recognisable by two-tape automata. We show that when such a semigroup happens to be a group, the converse actually holds, meaning quasi-automatic groups are asynchronously automatic.