Jumps in cohomology and free group actions

Abstract

A discrete group G has periodic cohomology over R if there is an element in a cohomology group cup product with which it induces an isomorphism in cohomology after a certain dimension. Adem and Smith showed that if R = Z, then this condition is equivalent to the existence of a finite dimensional free-G-CW-complex homotopy equivalent to a sphere. It has been conjectured by Olympia Talelli, that if G is also torsion-free then it must have finite cohomological dimension. In this paper we use the implied condition of jump cohomology over R to prove the conjecture for H F -groups and solvable groups. We also find necessary conditions for free and proper group actions on finite dimensional complexes homotopy equivalent to closed, orientable manifolds.

Action-induced nested classes of groups and jump (co)homology

“…In [26], we have shown that HL \ J Z D H 1 L. Combining this together with Proposition 4.6 implies HL \ H cell 1 .P Z / D H 1 L: (a) In Section 3, we pointed out that the group L @ !…”

“…In [26], we considered a new (co)homological condition for groups called jump (co)homology. In this section, we take a closer look at groups with this property.…”

“…In general, it has been conjectured that the first nine conditions on a group are equivalent (see [9,28,30]) and .2/ , .9/ has been verified for the class of LHF-groups and .1/ , .9/ for the class of all LHF-groups that have a bound on the order of their finite subgroups (see [30]). The equivalence .1/ , .10/ is known to hold for torsion-free LHF-groups (see [26,30]), and in [10], it was shown that any locally finite group (which is a group of jump height 0) of cardinality @ n has an .n C 1/-dimensional model for the classifying space for proper actions.…”

Spheroidal groups, virtual cohomology and lower dimensional G-spaces

“…I thank the referee for comments, and particularly for calling my attention to the two papers of N. Petrosyan which contain related results [3,Propsition 1.3] and [4,Theorem 4.2] which are proved in similar way. I thank J. Kollár for drawing my attention to such problems (see Section 3).…”

Cohomological invariants and the classifying space for proper actions

“…Until the recent work [ABJ 09], where groups with a strong global fixed point property are constructed, the only way to show that a group G did not belong to HF was to find a subgroup of G isomorphic to the Thompson group F. Here we show that certain branch groups, such as the first Grigorchuk group G is not contained in the class HF. Furthermore, G is a counterexample to a conjecture of Petrosyan [Pet07] and answers in negative a question of Jo-Nucinkis [JN08].…”