Symplectic Groups, Mapping Class Groups and the Stability of Bounded Cohomology

Mapping class groups satisfy cohomological stability. In this note we show how results by Bestvina and Fujiwara imply that the bounded cohomology does not stabilize, additionally we show that stabily polynomials in the Mumford-Morita-Miller classes are unbounded i.e. their norm tends to infinity as one increases the genus. While the bounded cohomology of the symplectic group does stabilize, we show that it does not stabilize via isometries in degree $2$. In order to establish this we calculate the norm of the signature class in $\mathrm{Sp}_{2h}(\mathbb{R})$ and estimate the norm of the integral signature class.