## Surface subgroups of non-positively curved groups III: Right-angled Artin groups

This is the third of three posts based on a guest lecture by Sam Kim.

In the last post, we considered the question of which Artin groups have hyperbolic surface subgroups. There we focused on Artin groups of finite and Euclidean type. Here, we turn to some non–positively curved (NPC) Artin groups, namely right–angled Artin groups (RAAGs).

A RAAG is an Artin group ${A(\Gamma,m)}$ in which all the edges of ${\Gamma}$ are labeled by ${2}$, and so has the presentation

$\displaystyle A(\Gamma) \ = \ \langle \, v\in V(\Gamma) \, | \, vw=wv \mbox{\ if } \{v,w\} \in E(\Gamma) \ \rangle.$

Charney and Davis found that RAAGs admit ${K(G,1)}$s that are NPC cube complexes. (A metric space is NPC if it’s universal cover is CAT(0)). This allows us to use the following theorems. [A flag complex is a simplicial complex ${\Delta}$ in which every complete subgraph of the 1-skeleton spans a simplex. Notice that if ${X}$ is a cube complex, then ${\mbox{Lk}(X,v)}$ is naturally a simplicial complex. An induced subcomplex is a maximal subcomplex with vertices in a given subset.]

Theorem 1 (Bridson–Haefliger, page 201). Let ${X}$ and ${Y}$ be complete, connected, geodesic metric spaces. If ${X}$ is NPC and ${f:Y\rightarrow X}$ is locally an isometric embedding, then ${f}$ induces an injective morphism ${f_*:\pi_1(Y) \rightarrow \pi_1(X)}$.

Theorem 2 (Bridson–Haefliger, page 212). Let ${X}$ be a finite dimensional cubical complex. Then ${X}$ is NPC if and only if ${\mbox{Lk}(X,v)}$ is a flag complex for every vertex ${v}$ of ${X}$.

Theorem 3 (Charney 2000). A cubical map ${f:Y \rightarrow X}$ between finite dimensional cubical complexes is a local isometric embedding iff the following conditions hold for every vertex ${v}$ of ${Y}$.

1. The induced map ${\mbox{Lk}(f,v):\mbox{Lk}(Y,v) \rightarrow \mbox{Lk}(X,f(v))}$ is injective.
2. The image of ${\mbox{Lk}(f,v)}$ is an induced subcomplex of ${\mbox{Lk}(X,f(v))}$.

Here are some partial results on the question of which RAAGs contain hyperbolic surface groups. [Notation: ${C_n}$ is a cycle on ${n}$ vertices. If ${\Gamma}$ is a graph, then ${\Gamma^*}$ is the graph with the same vertices as ${\Gamma}$ and exactly the edges that are not in ${\Gamma}$ — i.e. ${\{v,w \}}$ is an edge of ${\Gamma^*}$ if and only if it is not an edge of ${\Gamma}$.]

1. Droms–Servatius–Servatius${A(C_n)}$ contains a hyperbolic surface group for ${n\geq 5}$.
2. Crisp–Wiest — Every hyperbolic surface group with ${\chi < -1}$ embeds into some RAAG.
3. Kim${A(C_n^*)}$ contains hyperbolic surface subgroups for ${n \geq 5}$.
4. Crisp–Sageev–Sapir — Determined which ${A(\Gamma)}$ with ${|V(\Gamma)|\leq 8}$ contain hyperbolic surface subgroups.

The proofs of these results more–or–less follow the process of constructing a map from a surface into a ${K(G,1)}$ of a RAAG, and show that this map induces an injective map between the fundamental groups. The key difficulty lies in this latter step; however, thanks to the nonpositive curvature of the associated spaces, the three theorems quoted above apply.