Bin Sun
Michigan State University
Title
Cohomological dimensions of finitely generated groups and their subgroups
Abstract
I will present recent joint work with Francesco Fournier-Facio. Given a group G, consider the set S(G) of cohomological dimensions of its subgroups. We prove that, with obvious exceptions, every subset of N∪ {∞} can be realized as S(G) for some group G.
As an application, we answer a question of Talelli in the negative: there exists a torsion-free group G of infinite cohomological dimension such that all proper subgroups of G have uniformly bounded finite cohomological dimensions. Our construction also yields the first examples of torsion-free Smith groups–groups whose actions on CW-complexes always have global fixed points. Moreover, by analyzing the L2-Betti numbers of our examples, we obtain the first uncountable family of mutually non-measure equivalent, finitely generated, torsion-free groups.