\(L^2\)-determinant class and approximation of \(L^2\)-Betti numbers (Q2716138)

Let \(X\) be a finite CW-complex with fundamental group \(\pi\). The combinatorial Laplacians \(\Delta_p\) are positive selfadjoint operators on the spaces of square-summable \(p\)-cochains on the universal cover of \(X\), and may be viewed as elements of finite matrix rings over the von Neumann algebra of \(\pi\). The space \(S\) is said to be of \(\pi\)-determinant class if all the \(\Delta_p\) have finite regular Fuglede-Kadison determinant. It has been conjectured that every finite CW-complex is of determinant class. This was proven for residually finite groups by \textit{W. Lück} [Geom. Funct. Anal. 4, No. 4, 455-481 (1994; Zbl 0853.57021)] and for amenable groups by \textit{J. Dodziuk} and \textit{V. Mathai} [J. Funct. Anal. 154, No. 2, 359-378 (1998; Zbl 0936.57018)]. The main result of this paper is that the class of groups for which this is so is closed under direct and inverse limits, passing to subgroups and forming amenable extensions. (In particular it holds for all residually amenable groups). In other results it is shown that for spaces with fundamental groups in this class (i) the \(L^2\)-Reidemeister torsion is a homotopy invariant; (ii) if \(\pi\) has a nested sequence of subgroups with trivial intersection then the \(L^2\)-Betti numbers of \(X\) are the limits of the \(L^2\)-Betti numbers of the quotients of this sequence, acting on the associated intermediate covering spaces.