Bounded cohomology, quotient extensions, and hierarchical hyperbolicity

TL;DR

This work develops a cohomological and geometric framework linking bounded central extensions, quasihomomorphisms, and hierarchical hyperbolicity. It proves that for finitely generated abelian kernels, boundedness of a central extension is equivalent to the existence of a quasihomomorphism $\chi:E\to K$ that is the identity on $K$, and that bounded quotient extensions correspond to extendability of $\chi$ on the base group. A central theorem shows that if $G$ is HHG, then a central extension $1\to K\to E\to G\to 1$ is HHG if and only if the extension is bounded; under mild hypotheses with clean containers, the quotient $E'/K$ is commensurable to a HHG. These results illuminate when quotients of bounded central extensions remain bounded and tie the problem to extendability criteria and bounded cohomology in degree 3. The paper then applies the theory to quotients of mapping class groups, showing that quotients of the 4-strand braid group by suitable powers of a pseudo-Anosov are bounded central extensions of HHGs, and hence HHGs themselves, with implications for broader MCG quotients and conjectures on hierarchical hyperbolicity. The methods rely on quasimorphisms, Busemann constructions on hyperbolic domains, and rotating-family techniques to manage central actions, offering a pathway to analyze HHG-structure preservation under central quotients in related groups.

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.

Theorems & Definitions (53)

• Proposition 1.2: see Proposition \ref{['bounded iff extendable']}
• Theorem 1.3: see Theorem \ref{['thm:HHG_iff_bounded_abelian_ker']}
• Theorem 1.4: see Theorem \ref{['thm:centralquot_is_HHS']}
• Theorem 1.5: see Theorem \ref{['thm:braids']}
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Remark 2.5: Homogeneisation
• Definition 2.6
• Example 2.7: Busemann quasimorphism, see e.g. Manning_actions_on_hyp