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The higher connectivity at infinity of mapping class groups

Michael Mihalik

TL;DR

This work resolves the higher connectivity at infinity for a broad class of groups by proving a general criterion for simple connectivity at infinity when a finitely presented group contains a rank at least 3 free abelian subgroup with suitable commuting relations. Applying this to mapping class groups, it shows that $\Gamma_{g,0}^0$ with $g\ge 3$ are simply connected at infinity, and, as duality groups of dimension $4g-5$, are $4g-7$-connected at infinity via the Proper Hurewicz Theorem. The paper then provides a complete connectivity classification for all $\Gamma_{g,r}^s$, including precise end and semistability properties in exceptional cases. The methods combine combinatorial van Kampen diagram techniques with higher-dimensional cell attachments to push loops toward infinity, yielding explicit geometric and algebraic control over the asymptotic topology of mapping class groups.

Abstract

The higher connectivity at infinity for mapping class groups of surfaces with boundary components and punctures is understood with the exceptions of the mapping class groups for the closed surfaces of genus 3 and 4. In this paper we prove a general simply connected at infinity result for finitely presented groups that implies all mapping class groups of closed surfaces of genus $\geq 3$ are simply connected at infinity. As these groups are duality groups the Proper Hurewicz Theorem implies that they are $(n-2)$-connected at infinity where $n$ is the dimension of the group. Combining this result with earlier work we give a complete list of all mapping class groups and their connectivity at infinity.

The higher connectivity at infinity of mapping class groups

TL;DR

This work resolves the higher connectivity at infinity for a broad class of groups by proving a general criterion for simple connectivity at infinity when a finitely presented group contains a rank at least 3 free abelian subgroup with suitable commuting relations. Applying this to mapping class groups, it shows that with are simply connected at infinity, and, as duality groups of dimension , are -connected at infinity via the Proper Hurewicz Theorem. The paper then provides a complete connectivity classification for all , including precise end and semistability properties in exceptional cases. The methods combine combinatorial van Kampen diagram techniques with higher-dimensional cell attachments to push loops toward infinity, yielding explicit geometric and algebraic control over the asymptotic topology of mapping class groups.

Abstract

The higher connectivity at infinity for mapping class groups of surfaces with boundary components and punctures is understood with the exceptions of the mapping class groups for the closed surfaces of genus 3 and 4. In this paper we prove a general simply connected at infinity result for finitely presented groups that implies all mapping class groups of closed surfaces of genus are simply connected at infinity. As these groups are duality groups the Proper Hurewicz Theorem implies that they are -connected at infinity where is the dimension of the group. Combining this result with earlier work we give a complete list of all mapping class groups and their connectivity at infinity.
Paper Structure (6 sections, 11 theorems, 16 equations, 5 figures)

This paper contains 6 sections, 11 theorems, 16 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Simple Connectivity at Infinity and van Kampen Diagrams
  3. Principal Lemmas
  4. Outline of the Proof of the Main Theorem
  5. Proof of the Main Theorem
  6. Presentations for Mapping Class Groups

Key Result

Theorem 1.1

Suppose the group $G$ has a finite presentation $\langle S:R\rangle$ and $G$ contains a free abelian subgroup $A$ of rank $\geq 3$ with free generating set $T$ satisfying (1) and (2). Then $G$ is simply connected at infinity. (1) For $r\in R$ there is $t_r\in T$ such that $t_r$ commutes with each le

Figures (5)

  • Figure 1: Half Planes
  • Figure 2: Attaching and Mapping $Q|_{K_e}$
  • Figure 3: Adjusting Faces of 3-Cells
  • Figure 4: Double Attaching and Mapping
  • Figure 5: Adjusting vertex neighborhoods

Theorems & Definitions (12)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Remark 4.1
  • Theorem 5.1: Theorem 3, M4
  • Lemma 5.2
  • ...and 2 more