Surface Subgroups of Graph Groups

Abstract

Abstract.Given a graph T , define the group Fr to be that generated by the vertices of T, with a defining relation xy -yx for each pair x, y of adjacent vertices of T. In this article, we examine the groups Fr-, where the graph T is an H-gon, (n > 4). We use a covering space argument to prove that in this case, the commutator subgroup Ff contains the fundamental group of the orientable surface of genus 1 + (n -4)2"-3 . We then use this result to classify all finite graphs T for which Fp is a free group.To each graph T = ( V, E), with vertex set V and edge set E, we associate a presentation PT whose generators are the elements of V, and whose relations are {xy = yx\x ,y adjacent vertices of T).

On Right-Angled Artin Groups Whose Underlying Graphs Have Two Vertices With the Same Link

“…Motivated by the virtual Haken conjecture in 3-manifold theory, it has been an intriguing question to ask whether given a graph Γ the right-angled Artin group A(Γ) contains the fundamental group of a closed orientable hyperbolic surface or not ( [4,5,9,10,12]). For convenience, we will call the fundamental group of a closed orientable hyperbolic surface by a hyperbolic surface group.…”

“…H. Servatius, C. Droms, and B. Servatius showed that if Γ contains an induced n-cycle with n ≥ 5, called a long cycle, then A(Γ) contains a hyperbolic surface group ( [12]). S. Kim [9] showed that there is a graph Γ such that A(Γ) contains a hyperbolic surface group although Γ does not contain any long cycle.…”

Co-contractions of graphs and right-angled Artin groups

“…A./ contains a hyperbolic surface group, ie the fundamental group of a closed, hyperbolic surface, if there exists an induced C n for some n 5 in ; see Crisp and Wiest [3] and again Servatius, Droms and Servatius [19].…”

On Subgroups of the Pentagon Group

“…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 n−3 embeds into the graph group of the n-gon for n ≥ 4.…”

“…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 n−3 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.…”