The higher connectivity at infinity of mapping class groups

TL;DR

This paper proves that all mapping class groups of closed surfaces with genus at least 3 are simply connected at infinity, completing the understanding of their asymptotic topological properties.

Contribution

It establishes a general simply connected at infinity result for finitely presented groups, specifically applying to most mapping class groups of surfaces.

Findings

All mapping class groups of closed surfaces with genus ≥ 3 are simply connected at infinity.

Provides a complete list of mapping class groups and their connectivity at infinity.

Uses duality group properties and the Proper Hurewicz Theorem to derive connectivity results.

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.