On Subgroups of the Pentagon Group

Given a graph Γ with vertex set X, the graph group of Γ, denoted by G(Γ), is the group generated by the elements of X subject to the relations xy = yx for each pair x and y of adjacent vertices in Γ. In the literature graph groups are also known as partially commutative or right angled Artin groups. It has been known for some time that certain surface groups are embeddable into graph groups [4, 1]. In [4] it was shown that the fundamental group of a closed orientable surfaces of genus 1 + (n − 4)2 embeds into the graph group of the n-gon for n ≥ 4. Crisp and Wiest [1] deal with all surface groups which can possibly be embedded into a graph group and their embeddings are, in addition, quasi-isometric. The methods in these two papers are topological. The purpose of this note is to establish the following result which implies that the fundamental group of every closed orientable surface is embeddable into the graph group of the pentagon. This result appears to be new and its proof is fairly short and purely algebraic.