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Extensions of multicurve stabilizers are hierarchically hyperbolic

Jacob Russell

TL;DR

The paper proves that extensions of the surface group by stabilizers of multicurves, realized as full preimages $E_\alpha$ in $\operatorname{MCG}(S;z)$, are hierarchically hyperbolic groups. The approach builds a hyperbolic model $X_\alpha$ from the star of $\alpha$ and a pants-like graph $W_\alpha$, formulates a combinatorial HHS $(X_\alpha,W_\alpha)$, and then resolves annular issues by a blow-up construction $B(X_\alpha),B(W_\alpha)$ to obtain a metrically proper action; this yields an HHG structure for $E_\alpha$ with controlled geometry and projection data. The results answer a question of DDLS by showing HHG-extensions for multicurve stabilizers, and they yield concrete corollaries, such as a quadratic Dehn function, finite-order subgroup structure, and solvable conjugacy problems; the appendix extends Morse-boundary analysis to HHGs whose largest acylindrical action is on a quasi-tree, producing an $\omega$-Cantor Morse boundary in this setting. Overall, the work provides a robust geometric framework for understanding geometrically finite-like subgroups of the mapping class group through surface-group extensions and hierarchical hyperbolicity, with broad implications for stability, boundaries, and rigidity phenomena in related groups.

Abstract

For a closed and orientable surface of genus at least 2, we prove the surface group extensions of the stabilizers of multicurves 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 multicurve stabilizers are hierarchically hyperbolic

TL;DR

The paper proves that extensions of the surface group by stabilizers of multicurves, realized as full preimages in , are hierarchically hyperbolic groups. The approach builds a hyperbolic model from the star of and a pants-like graph , formulates a combinatorial HHS , and then resolves annular issues by a blow-up construction to obtain a metrically proper action; this yields an HHG structure for with controlled geometry and projection data. The results answer a question of DDLS by showing HHG-extensions for multicurve stabilizers, and they yield concrete corollaries, such as a quadratic Dehn function, finite-order subgroup structure, and solvable conjugacy problems; the appendix extends Morse-boundary analysis to HHGs whose largest acylindrical action is on a quasi-tree, producing an -Cantor Morse boundary in this setting. Overall, the work provides a robust geometric framework for understanding geometrically finite-like subgroups of the mapping class group through surface-group extensions and hierarchical hyperbolicity, with broad implications for stability, boundaries, and rigidity phenomena in related groups.

Abstract

For a closed and orientable surface of genus at least 2, we prove the surface group extensions of the stabilizers of multicurves 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.

Paper Structure

This paper contains 31 sections, 48 theorems, 20 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Geometric finiteness in $\operatorname{MCG}(S)$
  3. Outline of the paper
  4. Background and notation
  5. Simplicial Complexes
  6. Coarse geometry and hyperbolic spaces
  7. Groups acting on graphs
  8. Surfaces, curves, and mapping class groups
  9. Complexes of curves and subsurface projection
  10. Combinatorial HHSs
  11. A sketch of the proof
  12. Tree guided spaces
  13. A combinatorial HHS without annuli
  14. Definition of $X_\alpha$ and $W_\alpha$
  15. Combinatorial conditions for $(X_\alpha,W_\alpha)$
  16. ...and 16 more sections

Key Result

Theorem 1.1

Let $S$ be a closed orientable surface with genus at least 2. Let $\alpha$ be a multicurve on $S$ and $G_\alpha$ be the stabilizer of $\alpha$ in $\operatorname{MCG}(S)$. If $E_\alpha$ is the full preimage of $G_\alpha$ in $\operatorname{MCG}(S;z)$, then $E_\alpha$ is a hierarchically hyperbolic gro

Figures (4)

  • Figure 1: Replacing the gray curve in each picture with the black curve it intersects gives an example of the two different types of flip moves in the pants graph.
  • Figure 2: The surgery of a curve across a bigon containing the marked point.
  • Figure 3: An example of the subsurface $U_\mu$. The top picture shows the fixed multicurve $\alpha$ on the closed surface $S$. The second picture is of the multicurve $\mu \subseteq X_\alpha$ on the surface $S^z$ (where $\ast$ denotes the marked point $z$). The bottom picture is the subsurface $U_\mu$. The core curve of the annulus in the bottom picture is $\alpha_\mu$.
  • Figure 4: Schematic for the proof of Claim \ref{['claim:psi_is_independent_of_fiber']}

Theorems & Definitions (116)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Definition 2.1: Join, link, and star
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4: MMII
  • Definition 2.6: Modified curve graph
  • ...and 106 more