Approximating L^2 invariants of amenable covering spaces: A   combinatorial approach
  In this paper, we prove that the $L^2$ Betti numbers of an amenable covering space can be approximated by the average Betti numbers of a regular exhaustion, proving a conjecture that we made in an earlier paper. We also prove that an arbitrary amenable covering space of a finite simplicial complex is of determinant class. 
Jozef Dodziuk and Varghese Mathai