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A regular set of words is (k-)locally testable if mem- bership of a word in the set is determined by the nature of its sub- words of some bounded length k. In this article we study groups for which the set of all geodesic words with respect to some gener- ating set is (k-)locally testable, and we call such groups (k-)locally testable. We show that a group is 1-locally testable if and only if it is free abelian. We show that the class of (k-)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 com- pute 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.

GROUPS WHOSE GEODESICS ARE LOCALLY TESTABLE