Applications of \(L^2\) methods to infinite groups

Abstract

A finite presentation \(\mathcal{P}\) of a group \(G\) determines a finite 2-complex \( C(\mathcal{P}) \), with Euler characteristic \( \chi(C(\mathcal{P}))=1-g+r \), where \(g\) and \(r\) are the numbers of generators and relators of the presentation, respectively. The Euler characteristic of a finite complex is multiplicative under passage to finite covers, and is also the alternating sum of its Betti numbers. Less well known is that it is also the alternating sum of its \(L^2\)-Betti numbers, which are multiplicative under passage to finite covers (unlike the usual Betti numbers).

We shall sketch the definition of the \( L^2 \)-Betti numbers and show how they may be used to obtain strong results on groups with finite presentations of deficiency \( g-r>0 \). For instance, if \( G \) is such a group and the commutator subgroup \( G' \) is also finitely presentable then \( G' \) is free. With further work of Kochloukova, on Novikov extensions of group rings, it suffices to assume \( G' \) finitely generated.