Automaticity of non-positively curved $k$-fold triangle groups

TL;DR

This paper proves that non-positively curved $k$-fold triangle groups possess a regular language of geodesics and an automatic structure, enhancing understanding of their geometric and algebraic properties.

Contribution

It establishes the automaticity of non-positively curved $k$-fold triangle groups by showing they have finitely many cone types and a regular language of geodesics.

Findings

Finite cone types for these groups

Regular language of all geodesics

Automatic structure with fellow traveller property

Abstract

We show that non-positively curved $k$-fold triangle groups have finitely many cone types, and hence a regular language of all geodesics. Further, we prove that the language of lexicographically first geodesics is both regular and satisfies the fellow traveller property, giving an automatic structure for this family of groups.