Curve complex as a coset intersection complex

TL;DR

The paper proves that for any finitely generated group $H$ quasi-isometric to the mapping class group $\mathrm{Mod}(S)$ of a surface $S$, the curve complex $\mathcal{C}(S)$ is combinatorially equivalent to a coset intersection complex $\mathcal{K}(H,\mathcal{R})$ via the nerve, with $\mathcal{R}$ finite. It shows $\mathcal{N}(\mathcal{K}(H,\mathcal{R}))$ recovers $\mathcal{C}(S)$ and that $\mathrm{Aut}(\mathcal{K}(H,\mathcal{R}))$ is isomorphic to $\mathrm{Mod}^{\pm}(S)$, extending Ivanov’s metaconjecture to this broader class. The results hinge on quasi-isometric rigidity of $\mathrm{Mod}^{\pm}(S)$, the nerve of maximal simplices, and a nerve-homotopy framework to connect coarse geometry with classical surface topology. Additionally, the paper establishes a quasi-isometry and homotopy equivalence between $\mathcal{C}(S)$ and appropriate coset intersection complexes, and provides an equivariant embedding of $\mathcal{C}(S)$ as a subcomplex of a coset intersection complex, highlighting both the reach and limits of such combinatorial models.

Abstract

We show that, for every finitely generated group quasi-isometric to the mapping class group of a surface, there is a collection of subgroups such that their coset intersection complex is combinatorially equivalent to the curve complex, in the sense that one can be obtained from the other via taking a nerve. We also prove that the automorphism group of this coset intersection complex is the extended mapping class group, providing new evidence for Ivanov's metaconjecture.

Theorems & Definitions (71)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6: Ivanov, Luo
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9
• Definition 2.1