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

Abstract

Let $N_g$ be a closed non-orientable surface of genus $g\ge 3$, and let ${\mathrm{Homeo}}_0(N_g,\mu )$ be the identity component of its group of measure-preserving homeomorphisms. For every $n\ge 2$ we construct a linear injection ${\overline{EH}}_b^n(F_2)\hookrightarrow {\overline{EH}}_b^n({\mathrm{Homeo}}_0(N_g,\mu ))$, where ${\overline{EH}}_b^n$ denotes reduced exact bounded cohomology. In particular, $H_b^2({\mathrm{Homeo}}_0(N_g,\mu ))$ and $H_b^3({\mathrm{Homeo}}_0(N_g,\mu ))$ are infinite-dimensional. The main new issue is low genus. A natural figure-eight subgroup $F_2\le {\pi }_1(N_g)$ is hyperbolically embedded for $g\ge 4$, whereas in genus three hyperbolic embeddedness fails for every ${\pi }_1$-injective figure-eight with orientable regular neighborhood. We overcome this obstruction using the amalgam decomposition ${\pi }_1(N_g)=F_2{\ast }_{\mathbb{Z}}{F}_{g-2}$ 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.

22 pages, 4 figures