Groups whose geodesics are locally testable
Abstract
A regular set of words is (-)locally testable if membership of a word in the set is determined by the nature of its subwords of some bounded length . In this article we study groups for which the set of all geodesic words with respect to some generating set is (-)locally testable, and we call such groups (-)locally testable. We show that a group is \klt{1} if and only if it is free abelian. We show that the class of (-)locally testable groups is closed under taking finite direct products. We show also that a locally testable group has finitely many conjugacy classes of torsion elements. Our work involved computer investigations of specific groups, for which purpose we implemented an algorithm in \GAP\ to compute a finite state automaton with language equal to the set of all geodesics of a group (assuming that this language is regular), starting from a shortlex automatic structure. We provide a brief description of that algorithm.
Cite
@article{arxiv.1111.0775,
title = {Groups whose geodesics are locally testable},
author = {S. Hermiller and Derek F. Holt and Sarah Rees},
journal= {arXiv preprint arXiv:1111.0775},
year = {2011}
}