Torsion homology and cellular approximation

Abstract

In this note we describe the role of the Schur multiplier in the structure of the p-torsion of discrete groups. More concretely, we show how the knowledge of H2G allows to approximate many groups by colimits of copies of p-groups. Our examples include interesting families of non-commutative infinite groups, including Burnside groups, certain solvable groups and branch groups. We also provide a counterexample for a conjecture of E. Farjoun.

Generators and closed classes of groups

“…Moreover, mutatis mutandis, they may also be used to describe the Z/p-acyclic cover of these Burnside groups of high exponent. This is related with recent results about cellular covers of Burnside groups ( [25], [28] [22], [23]), that in particular have been useful to solve some conjectures by Farjoun [17]. On the other hand, note that we have approached the description of the acyclic and cellular covers of a non-nilpotent group with respect to an uncountable non-nilpotent group.…”

Cellular Covers of Local Groups

“…We rely on Ol'shanskiȋ's work in [17] where p > 10 10 , or rather on the improved bound p ≥ 665 in Adian and Atabekyan's [3]. Our next proposition is a particular case of the much more general statement [13,Proposition 3.9]. We identify cell p B by elementary methods for the sake of completeness, using in a crucial way the strong link between homotopical and group theoretical cellularization.…”

“…We have been wondering since then how to attack this problem for groups, one major obstruction being the difficulty to perform explicit computations. Our theorem is based on a very recent computation, [13], of a certain cellularization of large Burnside groups. The specific form of this cellularization is the key to the unexpected behaviour of the iteration of the two functors we choose.…”

On localizations of quasi-simple groups with given countable center

“…More related to our approach in this paper, (co)localizations of Burnside groups (which in turn are themselves localizations of free groups) have appeared at least in two different contexts: in relation with amenability phenomena [13], and from the point of view of combinatorial group theory, for example in [18] and [19]. In particular, the existence of 2 @ 0 varieties of groups not closed under cellular covers (another name for colocalizations) is shown in the last references, complementing the fact that there are 2 @ 0 varieties closed under taking cellular covers [14].…”