Rigidity of mapping class groups mod powers of twists

TL;DR

This paper investigates quotients of punctured sphere mapping class groups by large powers of Dehn twists, establishing automorphism rigidity, quasi-isometric rigidity, and small automorphism groups, extending Ivanov's theorems to these quotients.

Contribution

It introduces new techniques to analyze automorphisms and quasi-isometries of these quotients, proving rigidity results and characterizing their automorphism groups.

Findings

Automorphisms of quotients mirror those of original groups

Quotients exhibit quasi-isometric rigidity

Automorphism groups are small and well-understood

Abstract

We study quotients of mapping class groups of punctured spheres by suitable large powers of Dehn twists, showing an analogue of Ivanov's theorem for the automorphisms of the corresponding quotients of curve graphs. Then we use this result to prove quasi-isometric rigidity of these quotients, answering a question of Behrstock, Hagen, Martin, and Sisto in the case of punctured spheres. Finally, we show that the automorphism groups of our quotients of mapping class groups are "small", as are their abstract commensurators. This is again an analogue of a theorem of Ivanov about the automorphism group of the mapping class group. In the process we develop techniques to extract combinatorial data from a quasi-isometry of a hierarchically hyperbolic space, and use them to give a different proof of a result of Bowditch about quasi-isometric rigidity of pants graphs of punctured spheres.