## Notes of Henry Wilton’s Oxford lecture 21-03-2017

Surface subgrops of graphs of free groups with cyclic edge groups

1. Gromov’s questions

Tits’ Ping-pong Lemma. Let ${\Gamma}$ be a non-elementary hyperbolic group. If ${\alpha,\beta\in\Gamma}$ do not commute, then some high powers of ${\alpha}$ and ${\beta}$ generate a free subgroup.

Hence there exist heaps of free subgroups in hyperbolic groups. What about one-ended subgroups? Known: they willcome out in at most finitely many conjugacy classes. Need to rule out free groups, of course.

Question (Gromov). Does every one-ended hyperbolic group contain a surface 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.

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 ${BS(1,m)}$?

2. Free groups and relative questions

Relative means relative to a given set of relators. I.e. study group pairs ${(F,\omega)}$ where ${F}$ is e finitely generated free group and ${\omega}$ a finite collection of words. A surface pair is ${(\pi_1(\Sigma),\partial \Sigma)}$ where ${\Sigma}$ is a surface with boundary and ${\partial\Sigma=(c_1,\ldots,c_b)}$ represents its boundary components.

Note that a pair can be doubled into

$\displaystyle \begin{array}{rcl} D(\omega)=F\star_{\langle \omega\rangle} F. \end{array}$

Bestvina-Feighn: ${D(\omega)}$ is hyperbolic iff ${\omega}$ is not a proper power.

Shentzer: ${D(\omega)}$ is one-ended iff ${\omega}$ is not contained in a proper free factor. In this case, we say that the group pair is irreducible.

Theorem 1 (Wilton) Every irreducible group pair ${(F,\omega)}$ admits an essential admissible map of a surface with boundary ${(\pi_1(\Sigma),\partial \Sigma)}$.

Essential is a bit more restrictive than injective. Admissible means that along the boundary, the gluing map has the same degree on all components.

3. Consequences

3.1. Graphs of free groups

Corollary 2 For any one-ended graph of free groups with cyclic edge groups ${\Gamma}$,

1. either ${\Gamma}$ containes a Baumslag-Solitar subgroup,
2. or ${\Gamma}$ is hyperbolic and contains a surface subgroup.

There were earlier results with C. Gordon and Kim. In 2008, Calegari proved this under condition that ${H_2(\Gamma,{\mathbb Q})\not=0}$. He reduced the problem to linear equations, and the homology assumptions provided a solution. In 2010, Kim and Oun solved the case of doubles of rank 2 free groups, where Calegari’s equation could be scrutinized.

3.2. Residually free groups

A group is fully residually free if every finite subset is mapped injectively by some homomorphism to some free group.

Corollary 3 If ${L}$ is a finitely generated non-free fully residually free group, then ${L}$ contains a surface subgroup and ${\mathcal{C}(L)\not=\mathcal{C}(F)}$ for any free group ${F}$.

This follows from a structure theorem for such groups. This answers Remeslennikov’s question that arised in Alan Reid’s course.

3.3. Rigid groups

Say that a group is rigid if it does not split over 1 or ${{\mathbb Z}}$.

Theorem 4 (Louder-Touikan’s Strong Accessibility) If ${\Gamma}$ is a one-ended hyperbolic group, then ${\Gamma}$ cntains a quasi-convex subgroup ${H}$ such that

1. either ${H}$ is rigid,
2. or ${H}$ s a graph of free groups with cyclic edge-groups.

Hence Gromov’s question is reduced to the rigid case.

4. Ideas in the proof

1. Study all essential admissible maps ${(H,u)\rightarrow (F,\omega)}$ with ${(H,u)}$ irreducible. There is at least one, ${(F,\omega)}$ itslef.
2. Look at

$\displaystyle \begin{array}{rcl} \rho(H,u)=\frac{-\chi(H)}{n(u)}. \end{array}$

Argue that there is a map that maximizes ${\rho}$.

3. Show that this ${(H,u)}$ is a surface pair.

One can maybe implement the algorithm, but it does not seem to be efficient.

5. Comment

Arzhantseva: one cannot fix the genus of surfaces (1997).