On the dimension of classifying spaces for families of abelian subgroups

TL;DR

This paper establishes the minimal dimension of classifying spaces for finitely generated abelian groups with respect to families of subgroups based on torsion-free rank, providing a precise geometric characterization.

Contribution

It determines the exact dimension of classifying spaces for abelian groups relative to subgroups of bounded torsion-free rank, extending previous understanding of their geometric models.

Findings

Finitely generated abelian groups of rank n have an (n+r)-dimensional classifying space for subgroups of rank ≤ r.

The result generalizes known cases for torsion-free and finite subgroups.

Provides a framework for constructing minimal models for these classifying spaces.

Abstract

We show that a finitely generated abelian group $G$ of torsion-free rank $n\geq 1$ admits a $n+r$ dimensional model for the classifying space with isotropy in the family of subgroups of torsion-free rank less than or equal to $r\geq 0$.