Approximating \(\ell_2\)-Betti numbers of an amenable covering by ordinary Betti numbers
The author gives a simple homological proof of a recent result by \textit{J. Dodziuk} and \textit{V. Mathai} [J. Funct. Anal. 154, No. 2, 359-378 (1998; Zbl 0936.57018)] on the \(l_2\)-Betti numbers of an amenable covering of a finite cell complex. These \(l_2\)-Betti numbers [\textit{J. Cheeger} and \textit{M. Gromov}, Topology 25, 189-215 (1986; Zbl 0597.57020)], can be approximated by the ordinary Betti numbers \(\frac{\beta_p(Y_j)}{N_j}\) of a Følner exhaustion \(Y_j\) of the covering \(Y\), where \(N_j\) is the number of translates of a fundamental domain of \(Y\) necessary to cover \(Y_j\).
scientific article; zbMATH DE number 1278816