On the geometry of Cayley automatic groups

TL;DR

This paper investigates the geometric properties of Cayley automatic groups, introducing a distance function to measure their deviation from automaticity and establishing bounds for groups with certain algebraic properties.

Contribution

It demonstrates that for many non-automatic Cayley automatic groups, the deviation measure is linearly bounded, especially for groups with super-quadratic Dehn functions or not finitely presented.

Findings

The distance function is bounded below by a linear function for many non-automatic groups.

Groups with super-quadratic Dehn functions have a lower bound on the distance function.

Non-finitely presented Cayley automatic groups also exhibit a lower bound on the deviation measure.

Abstract

In contrast to being automatic, being Cayley automatic \emph{a priori} has no geometric consequences. Specifically, Cayley graphs of automatic groups enjoy a fellow traveler property. Here we study a distance function introduced by the first author and Trakuldit which aims to measure how far a Cayley automatic group is from being automatic, in terms of how badly the Cayley graph fails the fellow traveler property. The first author and Trakuldit showed that if it fails by at most a constant amount, then the group is in fact automatic. In this article we show that for a large class of non-automatic Cayley automatic groups this function is bounded below by a linear function in a precise sense defined herein. In fact, for all Cayley automatic groups which have super-quadratic Dehn function, or which are not finitely presented, we can construct a non-decreasing function which (1) depends only on the group and (2) bounds from below the distance function for any Cayley automatic structure on the group.