## Notes of Sebastian Hensel’s Cambridge lecture 01-03-2017

Rigidity and flexibility for handlebody groups

1. Rigidity

Our paradigm is Mostow rigidity: isomorphic cocompact lattices in ${Isom^+(H^n)}$ must be conjugate.

Consider mapping class groups, say in genus ${\geq 3}$ to avoid technicalities.

Theorem 1 (Ivanov) Isomorphic finite index subgroups of MCG are conjugate. In fact, MCG is its own commensurator.

There are many variants, for instance involving self maps of Teichmüller space.

Our goal is to study where MCG is replaced with a (underestimated) subgroup of MCG.

2. Handlebody groups

2.1. Definition

Let ${V}$ be a handlebody of genus ${g}$. We fix an identification of ${\partial V}$ with a surface ${\Sigma}$. Let ${H}$ be the subgroup of MCG of classes of homeos of ${\Sigma}$ which extend to homeos of ${V}$. It is isomorphic to the mapping class group of ${V}$ as a 3-manifold.

All handlebody groups are conjugate.

2.2. Crash course

Dehn twists along meridians (simple closed curves on ${\Sigma}$ that bound a disk in ${V}$) belong to ${H}$.

Dehn twists along simple closed curves that do not bound do not belong to ${H}$. Indeed, it maps meridians to non meridians. This shows that ${H}$ is not normal and has infinite index.

In the Torelli exact sequence

$\displaystyle \begin{array}{rcl} 1\rightarrow T\rightarrow MCG \rightarrow Sp(2g;{\mathbb Z})\rightarrow 1, \end{array}$

${H}$ is mapped to the stabilizer of a Lagrangian subspace ${L}$ in ${H_1(\Sigma,{\mathbb R})}$. Conversely, every symplectic matrix in ${Sp(2g;{\mathbb Z})}$ stabilizing ${L}$ is in the image of ${H}$ (Hirose).

It follows that there exist elements of the Torelli group ${T}$ which are not conjugated into ${H}$ (Jorgensen).

Beware that Dehn twists around meridians do not generate ${H}$. In fact they generate the kernel ${T'}$ of the following sequence (Luft):

$\displaystyle \begin{array}{rcl} 1\rightarrow T'\rightarrow H\rightarrow Out(F_g)\rightarrow 1. \end{array}$

3. Rigidity result

Theorem 2 (Hensel) Let ${\Gamma be a finite index subgroup. Any monomorphism ${f:\Gamma\rightarrow MCG_g}$ is conjugate to the standard embedding ${\Gamma\rightarrow H_g\rightarrow MCG_g}$.

Corollary 3 ${H}$ is its own commensurator.

Improves on Korhmaz-Schleimer: ${Out(H)=1}$.

4. Flexibility result

Super-rigidity seems hard. For instance, one does not know if a finite index subgroup of MCG can surject onto ${{\mathbb Z}}$.

Theorem 4 (Hensel) For every ${g}$, there exists ${h>g}$, a finite index subgroup ${\Gamma and a monomorphism ${f:\Gamma\rightarrow MCG_h}$ which is not conjugate into ${H_h}$.

These examples come from covers.

5. Proof of Ivanov’s theorem

Recall the scheme of proof of Mostow rigidity. Think of the given isomorphism ${\Gamma_1\rightarrow\Gamma_2}$ as a coarse geometric data: a quasi-isometry of hyperbolic space. This induces a boundary homeomorphism. Find extra symmetry (quasiconformality), show homeo is actually conformal. This is a candidate for extension into a hyperbolic isometry.

In our case, the coarse object is a simplicial automorphism of the curve graph. The extra symmetry. Show that such automorphisms are induced by elements of MCG.

5.1. Characterization of Dehn twists

Dehn twists have large centralizers, the centers of centralizers are small, so large (resp. small) that this characterizes powers of Dehn twists.

Lemma 5 (Ivanov) Isomorphisms between finite index subgroups of MCG map powers of Dehn twists to powers of Dehn twists.

It follows that an isomorphism defines an assignment curves to curves.

5.2. The curve graph

Its vertices correspond to isotopy classes of essential scimple closed curves. Edges correspond to disjointness (up to isotopy).

Since disjointness corresponds to commutation of Dehn twist, an isomorphism defines a simplicial map on the curve graph ${C(\Sigma)}$.

Theorem 6 ${Aut(C(\Sigma))=MCG(\Sigma)}$.

It follows that isomorphic is a conjugation.

6. Case of handlebody group

Something of centralizers and centers of centralizers remains.

Lemma 7 Monomorphisms between finite index subgroups of MCG map powers of Dehn twists around meridians to powers of Dehn twists.

Thus monomorphism induces an assignment meridians to curves, hence an superinjection of the disk graph ${D(V)}$ to ${C(\Sigma)}$.

Proposition 8 Any superinjection ${D(V_g)\rightarrow C(\Sigma_g)}$ is conjugate to the standard embedding.

This uses the fact that superinjections of ${C(\Sigma)}$ to itself are surjective.

This might seem to give us a control on the subgroup ${T'}$ only. No, elements of ${H}$ are indeed determined by their effect on meridians.

Unknown: does Torelli group ${T}$ have non-standard embeddings into MCG ?

7. Flexible examples

Take a meridian ${\alpha}$ on ${\Sigma}$, take a 3-fold cover. This extends to a branched cover between handlebodies, branched over a disk.