Bounded cohomology, quotient extensions, and hierarchical hyperbolicity

TL;DR

This paper investigates the properties of bounded central extensions of hierarchically hyperbolic groups (HHGs), demonstrating that such extensions preserve hierarchical hyperbolicity and exploring conditions under which quotients remain HHGs, with applications to braid and mapping class groups.

Contribution

It establishes that bounded central extensions of HHGs are themselves HHGs and characterizes when quotients of such extensions are also HHGs, linking to quasihomomorphism extendability.

Findings

Bounded central extensions of HHGs are HHGs.

Quotients of bounded central extensions are HHGs under certain conditions.

Quotients of specific braid groups are HHGs and bounded central extensions.

Abstract

We call a central extension bounded if its Euler class is represented by a bounded cocycle. We prove that a bounded central extension of a hierarchically hyperbolic group (HHG) is still a HHG; conversely if a central extension is a HHG, then the extension is bounded, and under a further mild assumption the quotient is commensurable to a HHG. Motivated by questions on hierarchical hyperbolicity of quotients of mapping class groups, we therefore consider the general problem of determining when a quotient of a bounded central extension is still bounded, which we prove to be equivalent to an extendability problem for quasihomomorphisms. Finally, we show that quotients of the 4-strands braid group by suitable powers of a pseudo-Anosov are HHG, and in fact bounded central extensions of some HHG. We also speculate on how to extend the previous result to all mapping class groups.