Measuring closeness between Cayley automatic groups and automatic groups

TL;DR

This paper introduces a numerical measure to estimate how close Cayley automatic groups are to being automatic, characterizes non-automatic Cayley automatic groups, and analyzes specific groups like lamplighter, Baumslag--Solitar, and Heisenberg groups.

Contribution

It proposes a new numerical characteristic to quantify the closeness of Cayley automatic groups to automatic groups and applies it to several important examples.

Findings

Characterizes non-automatic Cayley automatic groups using the new measure

Calculates the measure for lamplighter, Baumslag--Solitar, and Heisenberg groups

Provides insights into the structure of groups near the automatic boundary

Abstract

In this paper we introduce a way to estimate a level of closeness of Cayley automatic groups to the class of automatic groups using a certain numerical characteristic. We characterize Cayley automatic groups which are not automatic in terms of this numerical characteristic and then study it for the lamplighter group, the Baumslag--Solitar groups and the Heisenberg group.