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Spaces with vanishing \(\ell^2\)-homology and their fundamental groups (after Farber and Weinberger). (English) Zbl 0991.57002

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 \(k\geq 0\).

MSC:

57M07 Topological methods in group theory
55N35 Other homology theories in algebraic topology

Citations:

Zbl 0992.58012
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