Surface Subgroups of Right-Angled Artin Groups

Abstract

We consider the question of which right-angled Artin groups contain closed hyperbolic surface subgroups. It is known that a right-angled Artin group A(K) has such a subgroup if its defining graph K contains an n-hole (i.e. an induced cycle of length n) with n ≥ 5. We construct another eight "forbidden" graphs and show that every graph K on ≤ 8 vertices either contains one of our examples, or contains a hole of length ≥ 5, or has the property that A(K) does not contain hyperbolic closed surface subgroups. We also provide several sufficient conditions for a right angled Artin group to contain no hyperbolic surface subgroups.We prove that for one of these "forbidden" subgraphs P 2 (6), the right angled Artin group A(P 2 (6)) is a subgroup of a (right angled Artin) diagram group. Thus we show that a diagram group can contain a non-free hyperbolic subgroup answering a question of Guba and Sapir. We also show that fundamental groups of non-orientable surfaces can be subgroups of diagram groups. Thus the first integral homology of a subgroup of a diagram group can have torsion (all homology groups of all diagram groups are free Abelian by a result of Guba and Sapir).

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

“…It is not known whether the answer for this question is yes or no. In [4], Crisp, Sageev, and Sapir mentioned that the question can be answered affirmatively for one vertex amalgamation case, and they omitted a proof for that result. We give a detailed proof for one vertex amalgamation case, and use it to obtain the results regarding Question 1.1.…”

“…In Section 2, a part of the basic cutting lemma on dissections by Crisp, Sageev, and Sapir in [4] will be proved. In Section 3, we will discuss on Question 1.1 and Question 1.2.…”

“…In [5], for a finite set V , Crisp and Wiest introduced a V -dissection on a compact (orientable or non-orientable) surface (see [4,5,9,10]). Let V be a finite set.…”

“…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.…”

“…In [4], Crisp, Sageev, and Sapir found eight special graphs with less than or equal to 8 vertices and gave a description of a computer based proof that every graph Γ with less than or equal to 8 vertices contains either one of the eight special graphs or a long cycle as an induced subgraph if and only if A(Γ) contains a hyperbolic surface group.…”