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Rigidity of mapping class groups mod powers of twists

Giorgio Mangioni, Alessandro Sisto

TL;DR

The paper establishes rigidity phenomena for quotients of mapping class groups by large powers of Dehn twists on punctured spheres. It develops a lifting framework from quotient curve graphs to the curve graph, proves a combinatorial Ivanov-type rigidity for these quotients, and uses this to derive quasi-isometric rigidity and tight algebraic descriptions of automorphisms and commensurators. Central to the approach are projection and lifting lemmas, finite rigid sets, and a hinge-graph that translates quasi-isometries into automorphisms affecting standard flats. The results extend Ivanov-type rigidity to new Dehn-twist quotients, showing these quotients are quasi-isometrically and algebraically rigid and revealing their automorphism groups are small, with all quasi-isometries essentially induced by inner automorphisms.

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.

Rigidity of mapping class groups mod powers of twists

TL;DR

The paper establishes rigidity phenomena for quotients of mapping class groups by large powers of Dehn twists on punctured spheres. It develops a lifting framework from quotient curve graphs to the curve graph, proves a combinatorial Ivanov-type rigidity for these quotients, and uses this to derive quasi-isometric rigidity and tight algebraic descriptions of automorphisms and commensurators. Central to the approach are projection and lifting lemmas, finite rigid sets, and a hinge-graph that translates quasi-isometries into automorphisms affecting standard flats. The results extend Ivanov-type rigidity to new Dehn-twist quotients, showing these quotients are quasi-isometrically and algebraically rigid and revealing their automorphism groups are small, with all quasi-isometries essentially induced by inner automorphisms.

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.
Paper Structure (43 sections, 105 theorems, 33 equations, 30 figures)

This paper contains 43 sections, 105 theorems, 33 equations, 30 figures.

Table of Contents

  1. Introduction
  2. Open questions
  3. Outline of proofs
  4. Quasi-isometric rigidity
  5. Algebraic rigidity
  6. Outline of sections
  7. Setting and main tools
  8. Mapping classes and curves
  9. Complexity reduction and lifting properties
  10. Special intersections
  11. The finite rigid set
  12. Projection lemmas
  13. Isometric projections
  14. Puncture separations
  15. Lifting the rigid set
  16. ...and 28 more sections

Key Result

Theorem 1

Let $S=S_{0,b}$ be a punctured sphere, with $b\ge 7$ punctures. There exists $K_0\in\mathbb{N}_{>0}$ such that, if $K$ is a non-trivial multiple of $K_0$, then $H=MCG(S)/DT_K$ is quasi-isometrically rigid, meaning that if a finitely generated group $G$ is quasi-isometric to $H$ then $G$ and $H$ are

Figures (30)

  • Figure 1: The left twist $T$ of an annulus.
  • Figure 2: An example of a generalised hexagon. The dotted lines represent geodesics, while the triangles represent simplices.
  • Figure 3: The generalised pentagon $\mathfrak{P}$ described in \ref{['specint']}, which detects that any two non-adjacent vertices in the link of $R$ have special intersection.
  • Figure 4: The five curves involved in Definition \ref{['specint']} for $\Gamma=\mathcal{C}$, forming a chain on the five-holed sphere that $R$ cuts out. Any intersection is special, therefore the intersection number is always $2$.
  • Figure 5: Doubling these arcs gives a copy of $X_8$.
  • ...and 25 more figures

Theorems & Definitions (237)

  • Theorem 1: Quasi-isometric rigidity
  • Theorem 2: Algebraic rigidity
  • Theorem 3: Combinatorial rigidity
  • Remark 1
  • Remark 2
  • Definition 2.1: Dehn Twist
  • Definition 2.2: MM
  • Proposition 2.3
  • proof
  • Theorem 2.4: MM
  • ...and 227 more