Classifying spaces for families of virtually abelian subgroups of surface braid groups

TL;DR

This paper determines the minimal dimension of classifying spaces for families of virtually abelian subgroups in surface braid groups, linking it to virtual cohomological dimension and providing explicit calculations.

Contribution

It establishes the exact minimal dimension of classifying spaces for virtually abelian subgroups in surface braid groups, extending to the full braid group of the sphere and computing for amenable subgroups.

Findings

Minimal dimension equals virtual cohomological dimension plus n for certain surface braid groups.

Results apply to pure braid groups of surfaces with boundary or punctures.

Computed minimal models for the family of amenable subgroups in pure surface braid groups.

Abstract

Given a group $G$ and an integer $n \geq 0$, let $\mathcal{F}_n$ denote the family of all virtually abelian subgroups of $G$ of rank at most $n$. In this article, we show that for each $n \geq 1$, the minimal dimension of a model for the classifying space $E_{\mathcal{F}_n}G$ for the pure braid group of a surface of non-negative Euler characteristic with at least one boundary component or one puncture is equal to the virtual cohomological dimension of $G$ plus $n$. We prove an analogous result for the full braid group of the sphere. As an application, we compute the minimal dimension of a model for the classifying space associated to the family of amenable subgroups of pure surface braid groups.