Automaticity of non-positively curved $k$-fold triangle groups (2511.05141v1)

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.