On finitely generated submonoids of virtually free groups

TL;DR

This paper investigates finitely generated submonoids of virtually free groups, providing decision procedures, geometric characterizations, and solving the isomorphism problem, thereby extending understanding of their algebraic and computational properties.

Contribution

It introduces a new geometric characterization of these submonoids as quasi-geodesic monoids and solves the isomorphism problem for this class.

Findings

Decidability of whether a submonoid is graded

Equivalence of graded, regular, and Kleene monoids in free groups

Word problem for these monoids is rational

Abstract

We prove that it is decidable whether or not a finitely generated submonoid of a virtually free group is graded, introduce a new geometric characterization as quasi-geodesic monoids, and show that their word problem is rational (as a relation). We also solve the isomorphism problem for this class of monoids, generalizing earlier results for submonoids of free monoids. We also prove that the classes of graded monoids, regular monoids and Kleene monoids coincide for submonoids of free groups.