Dimensions of discrete groups and Brown's question Seminar

Nowadays, there are some very useful notions of a cohomological dimension of a group. Classically, for a torsion-free group G, the ordinary cohomological dimension cdG is equal to its geometric dimension gdG (the minimal dimension of a free contractible G-CW-complex) provided the cohomological dimension is not two. When cdG=2, the Eilenberg-Ganea Conjecture asserts that gdG=2.

For groups that contain torsion, the analogues of algebraic and geometric dimensions are less clear. This prompted K. S. Brown's question in 1977 for groups that are virtually torsion-free. It subsequently led to new notions of cohomological dimensions and to various other formulations of the Brown's question. I will discuss some history of this topic and the recent joint work with Ian Leary where we give the first counterexamples to the original Brown's question.