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Geometry of surface homeomorphism groups

Surfaces are a very interesting test case to study diffeomorphism and homeomorphism groups of dimension bigger than 1, where many more tools are available.

The study of surface symmetries naturally breaks into two parts:

$1 \to \mathrm{Homeo}_0(S) \to \mathrm{Homeo}^+(S) \to \mathrm{Mcg}(S) \to 1,$

where Homeo0(S) is a transformation group, and Mcg(S) is the mapping class group. These two groups have very different flavor:

• Mapping class groups are finitely generated, and thus can be studied using geometric group theory. Their algebra and geometry is by now fairly well understood: they are residually finite, hierarchically hyperbolic, algebraically and quasi-isometrically rigid, and much more.
• The group Homeo0(S) of isotopically trivial homeomorphisms is non-discrete, and are thus not accessible to (classical) geometric group theory. They are usually studied using geometric topology, and are generally much more mysterious.

They are algebraically simple, but almost nothing is known about their finitely generated subgroups, or their geometry (in the sense of Mann-Rosendal).

The underlying philosophy of this project is the transfer of methods which are successful for mapping class groups to the realm of homeomorphism groups, allowingto study the latter with geometric tools.

Our central goals can be grouped in the following subprojects:

• Lattices in homeomorphism groups, and elliptic or parabolic isometries of the Bowden-Hensel-Webb curve graph $$\mathcal{C}^\dagger(S)$$
• Stable Commutator Length and Hyperbolic Isometries of $$\mathcal{C}^\dagger(S)$$
• Large Scale Geometry of homeomorphism groups
• Nielsen Realisation Questions and mapping class group actions on $$\mathcal{C}^\dagger(S)$$.
• Extension problems for diffeomorphism groups and Automorphisms of $$\mathcal{C}^\dagger(S)$$.

## Publications

We show that the identity component of the group of diffeomorphisms of a closed oriented surface of positive genus admits many unbounded quasi-morphisms. As a corollary, we also deduce that this group is not uniformly perfect and its fragmentation norm is unbounded, answering a question of Burago--Ivanov--Polterovich. As a key tool we construct a hyperbolic graph on which these groups act, which is the analog of the curve graph for the mapping class group.

 Journal J. Amer. Math. Soc. Volume 35 Pages 211-231 Link to preprint version

Related project(s):
38Geometry of surface homeomorphism groups

Building on work of Bowden-Hensel-Webb, we study the action of the homeomorphism group of a surface $S$ on the fine curve graph $\mathcal{C}^\dagger(S)$.  While the definition of $\mathcal{C}^\dagger(S)$ parallels the classical curve graph for mapping class groups, we show that the dynamics of the action of $\mathrm{Homeo}(S)$ on $\mathcal{C}^\dagger(S)$ is much richer:  homeomorphisms induce parabolic isometries in addition to elliptics and hyperbolics, and all positive reals are realized as asymptotic translation lengths.

When the surface $S$ is a torus, we relate the dynamics of the action of a homeomorphism on $\mathcal{C}^\dagger(S)$ to the dynamics of its action on the torus via the classical theory of {\em rotation sets}.  We characterize homeomorphisms acting hyperbolically, show asymptotic translation length provides a lower bound for the area of the rotation set, and, while no characterisation purely in terms of rotation sets is possible, we give sufficient conditions for elements to be elliptic or parabolic.

 Journal preprint Link to preprint version

Related project(s):
38Geometry of surface homeomorphism groups

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