For a virtually torsion-free discrete group \(\Gamma\), a classifying space for proper actions (sometimes called a model for \(E \Gamma\)) is a \(\Gamma\)-CW-complex \(X\) such that stabilizers are finite and \(X^H\) is contractible for every finite subgroup \(H\) of \(\Gamma\). The minimal dimension of such a CW-complex is known as the geometric dimension \(\mathrm{gd}(\Gamma)\) of \(\Gamma\). It is natural to compare \(\mathrm{gd}(\Gamma)\) with other notions of dimension like the virtual cohomological dimension \(\mathrm{vcd}(\Gamma)\). It is easy to see that \(\mathrm{gd}(\Gamma) \geq \mathrm{vcd}(\Gamma)\) and there are examples (this is not easy) to show this inequality can be strict. Indeed, I. J. Leary and N. Petrosyan [Adv. Math. 311, 730–747 (2017; Zbl 1436.20093)] even came up with examples of \(\Gamma\) for which there is a model \(X\) as above with \(\Gamma \backslash X\) compact and such that \(\mathrm{gd}(\Gamma)> \mathrm{vcd}(\Gamma)\).
In the paper under review, the authors prove results on the positive side. They prove that for lattices in classical simple Lie groups, these dimensions coincide. The method of proof is to use the so-called Bredon cohomology. It is known that the Bredon cohomological dimension coincides with the geometric dimension except possibly when the former is \(2\) and the latter is \(3\). In this paper, the authors prove the main result by showing that the virtual cohomological dimension coincides with the Bredon cohomological dimension. Note that for torsion-free \(\Gamma\), they coincide any way and, therefore, the results of the paper are significant only when \(\Gamma\) has torsion.
Further, for certain groups like \(\mathrm{SL}(n,\mathbb{Z})\) and lattices in real-rank \(1\) simple Lie groups, the associated symmetric space admits a \(\Gamma\)-equivariant cocompact deformation retract \(X\) whose dimension coincides with \(\mathrm{vcd}(\Gamma)\). As the authors point out, this is unknown in the generality of all lattices as in the main theorem. However, they do deduce for the lattices in the main theorem that the associated symmetric space admits a \(\Gamma\)-equivariant proper \(\Gamma\)-CW-complex \(X\) homotopically equivalent to it such that dim \(X = \mathrm{vcd}(\Gamma\)). The authors work with the Borel-Serre compactification of the symmetric space.