Cohomological dimension with respect to perfect groups

Cohomological Approach to Asymptotic Dimension

We introduce the notion of asymptotic cohomology based on the bounded cohomology and define cohomological asymptotic dimension asdimZ X of metric spaces. We show that it agrees with the asymptotic dimension asdimX when the later is finite. Then we use this fact to construct an example of a metric space X of bounded geometry with finite asymptotic dimension for which asdim(X × R) = asdimX. In pa...

Automorphisms of Pro-p groups of finite virtual cohomological dimension

Let G be a pro-p group of finite cohomological dimension and type FP∞ and T is a finite p-group of automorphisms of G. We prove that the group of fixed points of T in G is again a pro-p group of type FP∞ (in particular it is finitely presented). Moreover we prove that a pro-p group G of type FP∞ and finite virtual cohomological dimension has finitely many conjugacy classes of finite subgroups.

Normal subgroups of profinite groups of finite cohomological dimension

We study a profinite group G of finite cohomological dimension with (topologically) finitely generated closed normal subgroup N . If G is pro-p and N is either free as a pro-p group or a Poincaré group of dimension 2 or analytic pro-p, we show that G/N has virtually finite cohomological dimension cd(G) − cd(N). Some other cases when G/N has virtually finite cohomological dimension are considere...

Non-manifold Hyperbolic Groups of High Cohomological Dimension

Virtual Cohomological Dimension of Mapping Class Groups of 3-manifolds

The mapping class group of a topological space is the group of self-homeomorphisms modulo the equivalence relation of isotopy. For 2-manifolds (of finite type), it is a discrete group which is known (see [M, HI, H2, H3, H4]) to share many of the properties of arithmetic subgroups of linear algebraic groups, although it is not arithmetic. In this note we describe the results of [Ml], which show ...