On the dimension of classifying spaces for families of abelian subgroups

Abstract

We show that a finitely generated abelian group G of torsionfree rank n 1 admits an (n + r)-dimensional model for E Fr G, where F r is the family of subgroups of torsion-free rank less than or equal to r 0. Definition 1.1. [5, (2.1)] Let F and G be families of subgroups of a given group G such that F ⊆ G. Let ∼ be an equivalence relation on G\F satisfying:

Classifying spaces for the family of virtually abelian subgroups of orientable 3-manifold groups

“…Let G be a virtually Z n group. In [Pry21, Proposition 1.3], it was proved that gd F k (G) ≤ n + k for 0 ≤ k < n. For a free abelian group this upper bound has also been proved in [CCMNP17]. In this article, we prove that this upper bound is sharp.…”

“…For n ≥ 2, the families F n have been recently studied by several people; see for example [Pry21,HP20,LASSn22,SSn20,JLS23]. For a virtually Z n group G it was proved in [Pry21] that gd F k (G) ≤ n + k for all 0 ≤ k < n. For a free abelian group this upper bound was also obtained by Corob-Cook, Moreno, Nucinkis and Pasini in [CCMNP17] and they asked whether this upper bound was sharp:…”

Classifying spaces for families of abelian subgroups of braid groups, RAAGs and graphs of abelian groups

“…In this subsection, we compute the F n -dimension of RAAGs and we give a lower bound for the F ngeometric dimension of the outer automorphism group of some RAAGs. We recall some basic notions about RAAGs, for further details see for instance [5]. Let be a finite simple graph, that is, a finite graph without loops or multiple edges between vertices.…”