On the geometry of Cayley automatic groups
3
Issued Date
2022-01-01
Resource Type
ISSN
02181967
Scopus ID
2-s2.0-85127081541
Journal Title
International Journal of Algebra and Computation
Rights Holder(s)
SCOPUS
Bibliographic Citation
International Journal of Algebra and Computation (2022)
Suggested Citation
Berdinsky D., Elder M., Taback J. On the geometry of Cayley automatic groups. International Journal of Algebra and Computation (2022). doi:10.1142/S0218196722500199 Retrieved from: https://repository.li.mahidol.ac.th/handle/123456789/87539
Title
On the geometry of Cayley automatic groups
Author(s)
Other Contributor(s)
Abstract
In contrast to being automatic, being Cayley automatic 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 paper, 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.