Take one compact cell complex (or manifold) X, and unroll to its universal cover. Grate the spectrum off of the L^2 Laplacian. Sprinkle liberally with von Neumann trace. Bake. *
I will give an overview of L^2 topological invariants, and then discuss Luck's theorem, which relates Betti numbers of finite covers of X to L^2 Betti numbers of X's universal cover. I will end with recent work that gives some understanding of growth rates of Betti numbers as one takes larger and larger finite covers of X.