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

TL;DR

This paper explores the contrasting behaviors of bounded cohomology in mapping class groups and symplectic groups, revealing non-stabilization in certain cases and providing explicit norm calculations for signature classes.

Contribution

It demonstrates that bounded cohomology does not stabilize for mapping class groups and shows non-stabilization via isometries in symplectic groups, with explicit norm computations.

Findings

Bounded cohomology of mapping class groups does not stabilize.

Stable polynomials in Mumford-Morita-Miller classes are unbounded.

Bounded cohomology of symplectic groups stabilizes but not via isometries in degree 2.

Abstract

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.