Surface groups in graphs of groups
1. Gromov’s questions
Today, all infinite groups are torsion-free.
Question (Gromov). Does every one-ended hyperbolic group contain a surface subgroup ?
Ping-pong produces heaps of free subgroups. However, one-ended subgroups arise in only finitely many sorts up to conjugacy, so it ought to be much harder to construct surface subgroups. Let us relax the question a bit: look for infinite index one-ended subgroups. One needs rule out surface group.
Question (Gromov). Does every one-ended hyperbolic group which is not a surface group, contain a finitely generated one-ended subgroup of infinite index?
Question. Does every one-ended finitely presented group contain either a surface group or a Baumslag-Solitar subgroup ?
The most important development is Kahn-Markovic’s positive result for 3-manifold groups. In 2012, Calegari-Walker gave a positive answer for random groups.
2. Free groups and relative questions
Here is a much easier class of examples.
Relative means relative to a given set of relators. I.e. study group pairs where is e finitely generated free group and a finite collection of words. A surface pair is where is a surface with boundary and represents its boundary components.
Note that a pair can be doubled into
Bestvina-Feighn: is hyperbolic iff is not a proper power.
Shentzer: is one-ended iff is not contained in a proper free factor. In this case, we say that the group pair is irreducible.
In 2008, Calegari proved this under condition that . He reduced the problem to linear equations, and the homology assumptions provided a solution.
There were earlier results with C. Gordon and Kim, we could treat infinite families of examples with .
In 2010, Kim and Oum solved the case of doubles of rank 2 free groups, where Calegari’s equation could be scrutinized. This solves question 1 in this special case.
I gave a complete affirmative answer to Question 1 for doubles in 2010.
Theorem 1 (Wilton) If and is hyperbolic, then it contains a surface subgroup.
Theorem 2 (Delzant-Potyagailo, Louder-Touikan) If is a one-ended hyperbolic group, then contains a quasi-convex subgroup such that
- either is connected without local cutpoints (rigid case).
- or satisfies the assumptions of the main theorem.
Hence for Gromov’s questions, one can reduce to the rigid case.
Corollary 3 Finite covers and linear algebra suffice to recognize primitive elements in free groups.
Corollary 4 Il is a finitely generated fully residually free group, and if for some free group , then . Finite covers and linear algebra suffice to recognize primitive elements in free groups.
3. Ideas in the proof
- Study all essential admissible maps with irreducible. There is at least one, itself.
- Define a cone defined by integral equations, whose integer points correspond to pairs of a surface with boundary and a boundary map.
- Look at
Argue that there is a map that maximizes .
- Show that this is a surface pair.
One can maybe implement the algorithm, but it does not seem to be efficient.