Hierarchically cocompact classifying spaces for mapping class groups of surfaces

TL;DR

This paper introduces hierarchically cocompact classifying spaces for mapping class groups of surfaces, establishing optimal bounds for their dimensions and answering a question posed by Lück.

Contribution

The authors define hierarchically cocompact classifying spaces and prove their existence with optimal bounds for mapping class groups of surfaces, extending understanding of their geometric properties.

Findings

Existence of hierarchically cocompact models for mapping class groups.

Optimal dimension bounds are established for closed surfaces.

Answers Lück's question on the dimension of classifying spaces.

Abstract

We define the notion of a hierarchically cocompact classifying space for a family of subgroups of a group. Our main application is to show that the mapping class group $\mbox{Mod}(S)$ of any connected oriented compact surface $S$, possibly with punctures and boundary components and with negative Euler characteristic has a hierarchically cocompact model for the family of virtually cyclic subgroups of dimension at most $\mbox{vcd} \mbox{Mod}(S)+1$. When the surface is closed, we prove that this bound is optimal. In particular, this answers a question of L\"{u}ck for mapping class groups of surfaces.