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 \(\mathcal{C}^\dagger(S)\).

• 1