Cayley automatic representations of wreath products

TL;DR

This paper develops automata-based representations of Cayley graphs for wreath products, providing bounds on element lengths and extending the concept of Cayley automatic groups.

Contribution

It introduces automata-based constructions for wreath product Cayley graphs aligned with existing notions of Cayley automatic groups and their extensions.

Findings

Automata representations for wreath product Cayley graphs

Bounds on element lengths in wreath products

Extension of Cayley automatic group concepts

Abstract

We construct the representations of Cayley graphs of wreath products using finite automata, pushdown automata and nested stack automata. These representations are in accordance with the notion of Cayley automatic groups introduced by Kharlampovich, Khoussainov and Miasnikov and its extensions introduced by Elder and Taback. We obtain the upper and lower bounds for a length of an element of a wreath product in terms of the representations constructed.