Torsion homology and cellular approximation

TL;DR

This paper explores how the Schur multiplier influences the p-torsion structure of discrete groups, enabling approximation of complex groups by colimits of finite p-groups, with applications to various infinite groups and a counterexample to a conjecture.

Contribution

It demonstrates the use of the Schur multiplier to approximate complex groups via colimits of finite p-groups, including new examples and a counterexample to a conjecture.

Findings

Approximation of infinite groups by finite p-groups using Schur multiplier.

Identification of specific families of groups where this approximation applies.

Counterexample to a conjecture of E. Farjoun.

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 $H_2G$ allows to approximate many groups by colimits of copies of finite $p$-groups. Our examples include interesting families of non-commutative infinite groups, including Burnside groups, certain solvable examples and the first Grigorchuk group. We also provide a counterexample for a conjecture of E. Farjoun.