** 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*.

** 2.1. History **

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.

** 2.2. Results **

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 3Finite covers and linear algebra suffice to recognize primitive elements in free groups.

Corollary 4Il 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.