On dimensions in Bredon homology

Abstract

We define a homological and cohomological dimension of groups in the context of Bredon homology and compare the two quantities. We apply this to describe the Bredon-homological dimension of nilpotent groups in terms of the Hirsch-rank. In particular this implies that for virtually torsion-free nilpotent groups the Bredon cohomological dimension is equal to the virtual cohomological dimension.

A note on the Mittag–Leffler condition for Bredon-modules

“…Observation 2.3. Using Yoneda type arguments analogously to those in [MV03,Nuc04] it follows directly that for any x P O F G the module dual to M " Rrx,´s F is M˚" Rr´, xs F and that the dual module of M " Rr´, xs F is M˚" Rrx,´s F .…”

“…In particular, the modules Rr´, ys F and Rrx,´s F are free, and any free is a direct sum of modules of this form. For detail the reader is referred to [Lüc89] for the general set-up, and to [Flu11] and [Nuc04] for this specific case. Cohomology and Homology functors ExtFpM, N q and Tor F pM, N q are now defined in the usual manner.…”

Cohomological Dimension of Mackey Functors for Infinite Groups

“…Can we conclude that also cd F G is finite? Question 6.1 is obviously true for polycyclic-by-finite groups and countable virtually torsion-free nilpotent groups as here vcdG = cd F G (see [11,17]).…”

Virtually soluble groups of type $\operatorname{FP}_∞$

“…Suppose cd N D 2. Then, by [23,Remark 6.4] it follows that N is finitely generated and hence, G is polycyclic-by-finite and has a 2-dimensional model for EG. Now suppose cd N D 1.…”