Bounded cohomology of measure-preserving homeomorphism groups of non-orientable surfaces

Abstract

Let $N_g$ be a closed non-orientable surface of genus $g\geq3$, and let $\operatorname{Homeo}_0(N_g,μ)$ be the identity component of its group of measure-preserving homeomorphisms. For every $n\geq2$ we construct a linear injection $\overline{EH}_b^n(F_2)\hookrightarrow\overline{EH}_b^n\bigl(\operatorname{Homeo}_0(N_g,μ)\bigr)$, where $\overline{EH}_b^n$ denotes reduced exact bounded cohomology. In particular, $H_b^2(\operatorname{Homeo}_0(N_g,μ))$ and $H_b^3(\operatorname{Homeo}_0(N_g,μ))$ are infinite-dimensional. The main new issue is low genus. A natural figure-eight subgroup $F_2\leqπ_1(N_g)$ is hyperbolically embedded for $g\geq4$, whereas in genus three hyperbolic embeddedness fails for every $π_1$-injective figure-eight with orientable regular neighborhood. We overcome this obstruction using the amalgam decomposition $π_1(N_g)=F_2*_{\mathbb Z}F_{g-2}$ and an isometric bounded-cohomology extension theorem for graphs of groups with amenable edge groups. Combined with the Gambaudo--Ghys transfer and annular pushing, this yields the injection above.