Towards quantitative classification of Cayley automatic groups

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.