Spectra of fractal groups and the Atiyah Conjecture

Rostislav Grigorchuk of Texas A&M University

After a quick introduction to the spectral theory of groups and graphs we take a more careful look at the so called lamplighter group L and show that the discrete Laplace operator on the Cayley graph of L (with respect to a certain generating set) is pure point spectrum and the spectral measure is discrete (and explicitly computed). This is the first example of a group with discrete spectral measure.

We take an unusual point of view and realize the lamplighter group L as a group generated by a 2-state automaton. This approach, along with some C* arguments, provides a crucial tool in our computations.

The above result is applied to answer a question of Michael Atyiah on the possible range of L^2 Betti numbers. Namely, we construct a 7 dimensional closed manifold whose third L^2 Betti number is not an integer (it is 1/3). The manifold also provides a counterexample to the so called strong Atiyah Conjecture concerning a relation between the range of L^2 Betti numbers and the orders of the finite subgroups of the fundamental group of the manifold.