A ug 2 00 0 APPROXIMATING L 2 TORSION ON AMENABLE COVERING SPACES

In this paper, we prove that the L combinatorial torsion of an amenable covering space can be approximated by the combinatorial torsions of a regular exhaustion. An ancillary theorem shows the L spectral density function of the combinatorial Laplacian on L-cochains on the covering space is approximated by the average spectral density functions of the combinatorial Laplacian on the cochains of the regular exhaustion, with either Dirichlet or Neumann boundary conditions, extending one of the main results in [DM]. The technique used incorporates some results of algebraic number theory. Introduction Consider a connected simplicial complex Y and a discrete amenable group π acting freely and simplicially on Y such that X = Y/π is a finite simplicial complex. Let F be a finite subcomplex of Y that is a fundamental domain of the π-action. The Følner criterion for amenability allows one to construct a regular exhaustion {Ym} ∞ m=1 of Y , a sequence of finite subcomplexes of Y satisfying the following conditions. 1. Ym consists of Nm translates g.F of F , g ∈ π. 2. Y = ∞