# Towards quantitative classification of Cayley automatic groups

## TL;DR

This paper introduces a numerical characteristic for Cayley automatic groups, explores its properties, and applies it to various group classes to facilitate their quantitative classification.

## Contribution

It formulates and proves properties of a new numerical characteristic for Cayley automatic groups, including its invariance and relationships with other group invariants.

## Key findings

- The numerical characteristic satisfies a fellow traveler property.
- It is invariant under finite extensions, direct products, and free products.
- The characteristic is studied for nilpotent groups, Heisenberg groups, and groups with exponential growth.

## Abstract

In this paper we address the problem of quantitative classification of Cayley automatic groups in terms of a certain numerical characteristic which we earlier introduced for this class of groups. For this numerical characteristic we formulate and prove a fellow traveler property, show its relationship with the Dehn function and prove its invariance with respect to taking finite extension, direct product and free product. We study this characteristic for nilpotent groups with a particular accent on the Heisenberg group, the fundamental groups of torus bundles over the circle and groups of exponential growth.