Surface subgroups of right-angled Artin groups

TL;DR

This paper investigates which right-angled Artin groups contain hyperbolic surface subgroups, identifying new forbidden graphs, providing conditions for absence of such subgroups, and linking certain RAAGs to diagram groups to answer longstanding questions.

Contribution

It introduces eight new forbidden graphs for hyperbolic surface subgroups in RAAGs and establishes connections between RAAGs and diagram groups, addressing open problems.

Findings

Identified eight forbidden graphs for hyperbolic surface subgroups in RAAGs.

Proved that certain RAAGs are subgroups of diagram groups, including non-orientable surface groups.

Showed that diagram groups can contain non-free hyperbolic subgroups and torsion in homology.

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 $n\geq 5$. We construct another eight "forbidden" graphs and show that every graph $K$ on $\le 8$ vertices either contains one of our examples, or contains a hole of length $\ge 5$, or has the property that $A(K)$ does not contain hyperbolic closed surface subgroups. We also provide several sufficient conditions for a \RAAG 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).