This is the first of three posts based on a guest lecture by Sam Kim.
In this post and the next we will consider the general question of which groups admit subgroups isomorphic to a surface group. We will focus on this problem for some particular classes of groups, namely Coxeter and Artin groups.
The virtual Haken conjecture
As motivation, we present the following famous conjecture of Waldhausen —
The Virtual Haken Conjecture (VHC). Every closed hyperbolic 3-manifold has a finite cover that contains an embedded, -injective, closed hyperbolic surface.
As proved by Scott, this problem can be divided into the following two steps. Let be a closed hyperbolic 3-manifold.
Step 1: Find a hyperbolic surface group in . (This amounts to showing that there is an immersed surface in .)
Step 2: Show that is LERF.
A group is LERF (locally extendable residually finite), if for any finitely generated subgroup and for all , there is a finite index subgroup such that .
Kahn and Markovic completed Step 1 in 2009. This provides us with a class of groups with hyperbolic surface subgroups. Step 2 remains open.
It appears that this approach to the VHC motivated the following question attributed to Gromov (see Wilton for comments on this attribution):
Question. Does every one–ended word–hyperbolic group contain a hyperbolic surface group?
One–ended groups are those that do not split as an amalgamated product or HNN extension over a finite subgroup. It suffices to answer the question for one–ended groups, since a surface subgroup of such an amalgamation or HNN extension is conjugate into one of the factors.
Surface subgroups of Coxeter groups
Coxeter groups are those with presentations
where is a simplicial graph with a labeling of its edges .
Notice that differs from the standard graph associated to a presentation of a Coxeter group in that two vertices are connected by an edge of label 2 in if they are not connected in the Coxeter diagram, and they are not connected in if they are joined by an edge labeled with in the Coxeter diagram.
Gromov’s question was resolved in the special case of Coxeter groups in 2004.
Theorem 1 (Gordon, Long and Reid). Every one–ended, word–hyperbolic Coxeter group contains a hyperbolic surface group.
In the remainder of this post we will sketch the proof of this theorem. We proceed via three lemmas.
Let be a cycle of length , and be the complete graph on vertices.
To prove this lemma, construct a hyperbolic polygon with sides, corresponding to the vertices of . (Exept when and all , then it will be an Euclidean square.) The angles between the sides correspond to the edges of , and are of size . Then is the group generated by the reflections across the sides of in the hyperbolic (or Euclidean) plane. Now, Selberg’s lemma states that has a finite index torsion-free subgroup. (A proof of Selberg’s lemma can be found here). This subgroup acts properly discontinuously and with compact quotient on hyperbolic space, by the action just constructed. So it is a surface group.
The next lemma can be proved by inducting on and using the classification of finite reflection groups.
Finally, we need a purely graph theoretical lemm proved by G. Dirac.
- is complete,
- splits over a complete graph, or
- contains an induced for some .
(An induced subgraph is a maximal subgraph with a given vertex set.)
Here is how the theorem follows from the lemmas. It is a standard result in the theory of Coxeter groups that if is an induced subgraph of , then naturally embeds into . If satisfies Condition 1 of Lemma 4, then we use Lemma 3. Suppose it satisfies Condition 2. Let be the complete subgraph over which splits. Then is an amalgamation over , and since we are assuming that is one-ended, must be infinite. Thus we use Lemma 3 again. Finally, if Condition 3 is satisfied, we use Lemma 2.
We have proved that contains a surface group. If we assume that is word–hyperbolic, then this subgroup must be a hyperbolic surface group.