Extensions of multicurve stabilizers are hierarchically hyperbolic

Abstract

For a closed and orientable surface S with genus at least 2, we prove the π 1 pSqextensions of the stabilizers of multicurves on S are hierarchically hyperbolic groups. This answers a question of Durham, Dowdall, Leininger, and Sisto. We also include an appendix that employs work of Charney, Cordes, and Sisto to characterize the Morse boundaries of hierarchically hyperbolic groups whose largest acylindrical action on a hyperbolic space is on a quasi-tree.

Extensions of Veech groups II: Hierarchical hyperbolicity and quasi-isometric rigidity

“…We note that the answer is 'yes' for the first example, since the extension group is the mapping class group of the surface S with a puncture. Moreover, since our work on this subject first appeared, Russell [55] addressed the second example by proving extensions of multicurve stabilizers are hierarchically hyperbolic groups.…”

Equivariant hierarchically hyperbolic structures for 3-manifold groups via quasimorphisms

“…This technique of building combinatorial HHSs by "blowing up the vertex groups" in some naturally-occurring hyperbolic graph is quite flexible, and has analogues in a number of other contexts. For example, it is applied in the context of certain Artin groups in [HMS21], extensions of lattice Veech groups in [DDLS20], and extensions of multicurve stabilisers in [Rus21]. In [BHMS20], it is explained how to build combinatorial HHSs for right-angled Artin groups and mapping class groups by respectively blowing up the Kim-Koberda extension graph [KK13] and the curve graph.…”

Extensions of Veech groups II: Hierarchical hyperbolicity and quasi-isometric rigidity

“…We note that the answer is 'yes' for the first example, since the extension group is the mapping class group of the surface S with a puncture. Moreover, since our work on this subject first appeared, Russell [Rus21] addressed the second example by proving extensions of multicurve stabilizers are hierarchically hyperbolic groups.…”