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 in which all the edges of are labeled by , and so has the presentation
Charney and Davis found that RAAGs admit 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 in which every complete subgraph of the 1-skeleton spans a simplex. Notice that if is a cube complex, then 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 and be complete, connected, geodesic metric spaces. If is NPC and is locally an isometric embedding, then induces an injective morphism .
Theorem 2 (Bridson–Haefliger, page 212). Let be a finite dimensional cubical complex. Then is NPC if and only if is a flag complex for every vertex of .
Theorem 3 (Charney 2000). A cubical map between finite dimensional cubical complexes is a local isometric embedding iff the following conditions hold for every vertex of .
- The induced map is injective.
- The image of is an induced subcomplex of .
Here are some partial results on the question of which RAAGs contain hyperbolic surface groups. [Notation: is a cycle on vertices. If is a graph, then is the graph with the same vertices as and exactly the edges that are not in — i.e. is an edge of if and only if it is not an edge of .]
- Droms–Servatius–Servatius — contains a hyperbolic surface group for .
- Crisp–Wiest — Every hyperbolic surface group with embeds into some RAAG.
- Kim — contains hyperbolic surface subgroups for .
- Crisp–Sageev–Sapir — Determined which with contain hyperbolic surface subgroups.
The proofs of these results more–or–less follow the process of constructing a map from a surface into a 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.