Groups whose geodesics are locally testable

TL;DR

This paper characterizes groups whose geodesic words are determined by local subword patterns, showing that free abelian groups are exactly those with 1-local testability and exploring properties of such groups.

Contribution

It introduces the concept of locally testable groups based on geodesic words, characterizes 1-local testability, and demonstrates closure properties and torsion element finiteness.

Findings

A group is 1-locally testable iff it is free abelian.

The class of locally testable groups is closed under finite direct products.

Locally testable groups have finitely many conjugacy classes of torsion elements.

Abstract

A regular set of words is ($k$-)locally testable if membership of a word in the set is determined by the nature of its subwords of some bounded length $k$. In this article we study groups for which the set of all geodesic words with respect to some generating set is ($k$-)locally testable, and we call such groups ($k$-)locally testable. We show that a group is \klt{1} 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 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.