[Paper Review] Classifying spaces for proper actions of mapping class groups

Guido Mislin|ArXiv.org|May 6, 2009

Geometric and Algebraic Topology19 references19 citations

TL;DR

This paper constructs cocompact models for the classifying spaces for proper actions of mapping class groups of oriented surfaces with genus $g$, $r$ boundary components, and $s$ punctures. Using induction on $r$ and $s$, and building on a known cocompact model for the case $\Gamma_{g,0}^0$, the author establishes the existence of such models for all $\Gamma_{g,r}^s$ by leveraging group extensions and geometric actions on symmetric spaces or hyperbolic planes.

ABSTRACT

We describe a cocompact model for the classifying space for proper actions of the mapping class group of a surface with punctures and boundary components. Our construction relies on a known model for the case of a closed surface and uses an induction on the number of punctures and boundary components.

Motivation & Objective

• To provide a uniform construction of cocompact models for the classifying spaces $\underline{E}\Gamma^{s}_{g,r}$ for all mapping class groups of oriented surfaces with genus $g$, $r$ boundary components, and $s$ punctures.
• To extend known results for the case $\Gamma_{g,0}^0$ to all configurations of $r$ and $s$ using inductive techniques.
• To establish the existence of proper, cocompact, and $\Gamma$-equivariant actions on contractible spaces for all such mapping class groups.
• To resolve the classification of proper actions for mapping class groups in low-genus cases ($g=0,1$) by leveraging known triviality and isomorphism results.
• To unify and generalize prior constructions, including those by Ji and Wolpert for $\Gamma_{g,0}^s$, into a single inductive framework.

Proposed method

• The construction proceeds by induction on the number of boundary components $r$ and punctures $s$, starting from the base case $\Gamma_{g,0}^0$.
• The inductive step uses exact sequences of the form $1 \to H \to \Gamma \to F \to 1$, where $F$ is a finite group and $H$ is either a finitely generated free group or a surface group $\pi_1(S_g)$ with $g>0$.
• For each such extension, the existence of a cocompact $\underline{E}\Gamma$ is established using geometric actions: on $\mathbb{R}^n$ for free groups and on the hyperbolic plane $U$ for surface groups with $g>1$.
• The key technique involves lifting finite group actions from the surface to its universal cover, using Kerckhoff’s theorem to ensure faithful isometric actions on the hyperbolic plane.
• The construction relies on the fact that finite extensions of groups with cocompact $\underline{E}\Gamma$ also admit such models, provided the kernel is a surface group or free group.
• The method applies the general principle that if $H$ admits a cocompact $\underline{E}H$ and $F$ is finite, then $\Gamma$ as an extension inherits a cocompact $\underline{E}\Gamma$ via geometric or algebraic lifting.

Experimental results

Research questions

• RQ1 Can a uniform construction of cocompact models for $\underline{E}\Gamma^{s}_{g,r}$ be given for all mapping class groups of oriented surfaces with $g$ genus, $r$ boundary components, and $s$ punctures?
• RQ2 How can the existence of cocompact models be extended from the base case $\Gamma_{g,0}^0$ to all $\Gamma_{g,r}^s$ using inductive methods?
• RQ3 What role do group extensions with finite quotients and surface or free group kernels play in constructing proper, cocompact $\Gamma$-CW-complexes?
• RQ4 How do geometric actions on symmetric spaces (e.g., $\mathbb{R}^n$ or the upper half-plane) ensure the existence of cocompact classifying spaces for proper actions?
• RQ5 What is the behavior of the construction in low-genus cases ($g=0,1$), and how are exceptional cases handled?

Key findings

• A cocompact model for $\underline{E}\Gamma^{s}_{g,r}$ exists for all mapping class groups $\Gamma_{g,r}^s$ of oriented surfaces with genus $g$, $r$ boundary components, and $s$ punctures.
• The construction is inductive: starting from $\Gamma_{g,0}^0$, which admits a cocompact model by prior work, the result extends to $\Gamma_{g,0}^s$ using the exact sequence (B), and then to $\Gamma_{g,r}^s$ using (D).
• For $g \geq 2$, the group $\Gamma_{g,r}^s$ admits a two-dimensional cocompact $\underline{E}\Gamma$ modeled on the hyperbolic plane $U$, with action by hyperbolic isometries.
• For $g=0$, the mapping class groups $\Gamma_{0,0}^s$ are trivial for $s < 4$, and the construction proceeds via exact sequences to extend to higher $s$ and $r$.
• The case $g=1$ is handled by noting $\Gamma_{1,0}^0 \cong \Gamma_{1,0}^1$, and extending via the same inductive framework.
• The method confirms that finite extensions of groups with cocompact $\underline{E}\Gamma$ also admit such models when the kernel is a surface group or finitely generated free group.