Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An EZ-structure for the mapping class group

Ursula Hamenstädt

TL;DR

This work constructs an explicit EZ-boundary for the mapping class group $\mathrm{Mod}(S)$ of a finite-type surface $S$ by augmenting the thick Teichmüller space $\mathcal{T}_\varepsilon(S)$ with a geometrically defined boundary ${\mathcal{X}}(S)$. The boundary ${\mathcal{X}}(S)$ is endowed with a carefully designed topology, built from subsurface projections and joins of curve-graph boundaries, making it a compact $\mathrm{Mod}(S)$-space on which the action is minimal, strongly proximal and topologically free. The pair $(\overline{\mathcal{T}}(S),{\mathcal{X}}(S))$ is shown to be an ${\mathcal{E\mathcal{Z}}}$-structure, with ${\mathcal{X}}(S)$ serving as the geometric boundary; this yields established consequences such as the Novikov conjecture and the Farrell–Jones conjecture for $\mathrm{Mod}(S)$. The construction unifies a Tits-type boundary ${\mathcal{X}}_T(S)$ via the oriented curve complex with a compactification framework built from Teichmüller geometry, subsurface projections, and CAT(0)-style boundary behavior, while providing explicit neighborhood bases and finite-dimensional control. This advances understanding of the large-scale geometry of $\mathrm{Mod}(S)$ and supports the conjectured equality between asdim and vcd for these groups.

Abstract

We construct a boundary for the mapping class group Mod(S) of a surface S of finite type. The action of Mod(S) on this boundary is minimal, strongly proximal and topologically free. The boundary is the boundary of an EZ-structure for Mod(S).

An EZ-structure for the mapping class group

TL;DR

This work constructs an explicit EZ-boundary for the mapping class group of a finite-type surface by augmenting the thick Teichmüller space with a geometrically defined boundary . The boundary is endowed with a carefully designed topology, built from subsurface projections and joins of curve-graph boundaries, making it a compact -space on which the action is minimal, strongly proximal and topologically free. The pair is shown to be an -structure, with serving as the geometric boundary; this yields established consequences such as the Novikov conjecture and the Farrell–Jones conjecture for . The construction unifies a Tits-type boundary via the oriented curve complex with a compactification framework built from Teichmüller geometry, subsurface projections, and CAT(0)-style boundary behavior, while providing explicit neighborhood bases and finite-dimensional control. This advances understanding of the large-scale geometry of and supports the conjectured equality between asdim and vcd for these groups.

Abstract

We construct a boundary for the mapping class group Mod(S) of a surface S of finite type. The action of Mod(S) on this boundary is minimal, strongly proximal and topologically free. The boundary is the boundary of an EZ-structure for Mod(S).

Paper Structure

This paper contains 14 sections, 44 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. The Tits boundary of ${\rm Mod}(S)$
  3. A topology for ${\mathcal{X}}(S)$
  4. Convergence to a minimal filling lamination
  5. Product spaces
  6. Projections and a topology on ${\mathcal{X}}(S)$
  7. A small boundary for ${\rm Mod}(S)$
  8. The thick part of Teichmüller space
  9. A topology on ${\mathcal{T}}_\epsilon(S)\cup {\mathcal{X}}(S)$
  10. Neighborhood bases
  11. A neighborhood basis for minimal filling laminations
  12. Neighborhoods of minimal filling laminations for disconnected surfaces
  13. Neighborhoods of arbitrary points
  14. Proof of the main Theorem

Key Result

Theorem 1

For sufficiently small $\epsilon <\epsilon_0$, the set ${\mathcal{T}}_\epsilon(S)$ is a manifold with corners which is a deformation retract of ${\mathcal{T}}(S)$. The mapping class group ${\rm Mod}(S)$ acts on ${\mathcal{T}}_\epsilon(S)$ properly and cocompactly.

Theorems & Definitions (100)

  • Theorem 1: Ji-Wolpert
  • Definition 2: Small boundary
  • Definition 3: ${\mathcal{E}\mathcal{Z}}$-structure
  • Theorem 4
  • Corollary 5
  • Corollary 6
  • Corollary 7
  • Proposition 8
  • Example 2.1
  • Remark 2.2
  • ...and 90 more