Star-free geodesic languages for groups

TL;DR

This paper proves that certain groups with specific small cancellation properties have geodesic languages that are star-free regular, and explores how this property depends on the generating set and group operations.

Contribution

It establishes that groups satisfying particular small cancellation conditions have star-free geodesic languages and shows the property is preserved under group products.

Findings

Groups with $C'(1/6)$ or $C'(1/4)-T(4)$ conditions have star-free geodesic languages.

Star-free property depends on the generating set, even in free groups.

Closure of star-free geodesic groups under graph and direct products, including virtually abelian groups.

Abstract

In this article we show that every group with a finite presentation satisfying one or both of the small cancellation conditions $C'(1/6)$ and $C'(1/4)-T(4)$ has the property that the set of all geodesics (over the same generating set) is a star-free regular language. Star-free regularity of the geodesic set is shown to be dependent on the generating set chosen, even for free groups. We also show that the class of groups whose geodesic sets are star-free with respect to some generating set is closed under taking graph (and hence free and direct) products, and includes all virtually abelian groups.