Higson, Nigel; Roe, John; Schick, Thomas Spaces with vanishing \(\ell^2\)-homology and their fundamental groups (after Farber and Weinberger). (English) Zbl 0991.57002 Geom. Dedicata 87, No. 1-3, 335-343 (2001). In this elegant paper the authors completely characterize those groups which can occur as fundamental groups of finite CW-complexes with vanishing \(\ell^2\)-homology. The first examples of such groups were obtained by M. Farber and S. Weinberger [On the zero-in-the-spectrum conjecture, Ann. Math. (2) 154, No. 1, 139-154 (2001; Zbl 0992.58012)]. The main result of the paper is: Let \(G\) be a finitely presented group and suppose that the homology groups \(H_k(G,\ell^2(G))\) are zero for \(k=0, 1, 2.\) For every dimension \(n\geq 6\) there is a closed manifold \(M\) of dimension \(n\) with \(\pi_1(M)=G\) and such that \(H_k(M,\ell^2(G))=0\) for all \(k\geq 0\). Reviewer: M.Farber (Tel Aviv) Cited in 2 ReviewsCited in 7 Documents MSC: 57M07 Topological methods in group theory 55N35 Other homology theories in algebraic topology Keywords:\(\ell^2\)-homology; \(L^2\)-cohomology; zero-in-the-spectrum conjecture Citations:Zbl 0992.58012 PDFBibTeX XMLCite \textit{N. Higson} et al., Geom. Dedicata 87, No. 1--3, 335--343 (2001; Zbl 0991.57002) Full Text: DOI arXiv