Bredon cohomological dimensions for groups acting on $\mathrm{CAT}(0)$-spaces

Abstract

Let G be a group acting isometrically with discrete orbits on a separable complete CAT(0)-space of bounded topological dimension. Under certain conditions, we give upper bounds for the Bredon cohomological dimension of G for the families of finite and virtually cyclic subgroups. As an application, we prove that the mapping class group of any closed, connected, and orientable surface of genus g ≥ 2 admits a (9g − 8)-dimensional classifying space with virtually cyclic stabilizers. In addition, our results apply to fundamental groups of graphs of groups and groups acting on Euclidean buildings. In particular, we show that all finitely generated linear groups of positive characteristic have a finite dimensional classifying space for proper actions and a finite dimensional classifying space for the family of virtually cyclic subgroups. We also show that every generalized Baumslag-Solitar group has a 3-dimensional model for the classifying space with virtually cyclic stabilizers.

Hierarchically cocompact classifying spaces for mapping class groups of surfaces

“…If , then . So, by [, Lemma 5.2], . Combining this with [, Proposition 9] shows that .…”

“…Finite dimensional models for EMod(S) have been exhibited by Degrijse and the second author in [5] for closed surfaces S (the obtained bound on dimension is 9g − 8) and later by Juan-Pineda and Trujillo-Negrete in [13] for surfaces S that have negative Euler characteristic with possible punctures and boundary components (the obtained bound is [Mod(S) : Mod(S)[m]](vcd Mod(S) + 1), m 3 where Mod(S)[m] is the level m congruence subgroup). Apart from exhibiting models that are mapping telescopes of cocompact models, our bounds substantially improve on the bounds given there.…”

Complete Bredon cohomology and its applications to hierarchically defined groups

“…Suppose H is a finitely generated subgroup of G. If G is a subgroup of GL n (C) of integral characteristic, then by [1] when F is the class of finite groups or or by the previous theorem when F is the class of virtually cyclic groups, we know that H in H F F. If G embeds into GL n (F) for some field F of positive characteristic, then by [7,Corollary 5], H has finite Bredon cohomological dimension and hence it is in H F F. This shows that G is in LH F F. The result now follows from Theorem 4.4.…”

“…Finding manageable models for EG, the classifying space for virtually cyclic isotropy, has been shown to be much more elusive. So far manageable models have been found for crystallographic groups [17], polycyclic-by-finite groups [24], hyperbolic groups [12], certain HNN-extensions [10], elementary amenable groups of finite Hirsch length [5,6,11] and groups acting isometrically with discrete orbits on separable complete CAT(0)-spaces [7,22]. Let F be a family of subgroups of a given group and denote by E F G the classifying space with isotropy in F. In this note we propose a method to decide whether a group has a finite dimensional model for E F G without actually providing a bound.…”

“…If F is the class of all finite groups, then LH F F also turns out to be quite large. It contains all elementary amenable groups and all linear groups over a field of arbitrary characteristic (see [7], [8]). It is also closed under extensions, taking subgroups, amalgamated products, HNN-extensions, and countable directed unions.…”