In this paper we also consider L2 -torsion. At firrt, L2-torsion was defined for L2 - acyclic covering spaces. The L2 -analytic torsion was first studied in [18] and [14], and L2-Reidemeister-Franz torsion was first studied in [6] (see also [16] ). Equal- ity of the combinatorial and analytic L2-torsions was proven in 1996 [4]. In order to define these L2-torsions, one needs to establish decay near zero of the spectral density function for the L2-Laplacian. In the case of a residually finite covering, Luck [15] derives an elegant estimate on the spectral density functions for the finite covers that in the limit gives the necessary decay for the combinato- rial L2 -Laplacian. Lilck also proves the homotopy invariance of L2 -combinatorial torsion in this case.
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