  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.
