In this more technical talk I'll try to describe in more detail the theorems from mathematical physics which provide the difficult part of the proofs of the theorems from the previous talk, i.e. that 1) Novikov-Shubin invariants can be 0, and 2) for a given group, depending on the generating set, the spectral measure of the random walk operator can be either pure-point or singularly continuous. If time permits, I will talk about the random walk operators on the discrete Heisenberg group where answering analogous questions seems to be much more difficult.

*Łukasz Grabowski, University of Oxford *