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).

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About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
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