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Department of Mathematics,
University of California San Diego

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Special Colloquium

Rostislav Grigorchuk

Texas A&M

Spectra of fractal groups and the Atiyah Conjecture

Abstract:

After a quick introduction to the spectral theory of groups and graphswe take a more careful look at the so called lamplighter group Land show that the discrete Laplace operator on the Cayley graph of L(with respect to a certain generating set) is pure point spectrum andthe spectral measure is discrete (and explicitly computed). This is thefirst example of a group with discrete spectral measure.We take an unusual point of view and realize the lamplighter group L asa group generated by a 2-state automaton. This approach, along with someC* arguments, provides a crucial tool in our computations.The above result is applied to answer a question of Michael Atyiah onthe possible range of $L^2$ Betti numbers. Namely, we construct a 7dimensional closed manifold whose third $L^2$ Betti number is not aninteger (it is 1/3). The manifold also provides a counterexample to theso called strong Atiyah Conjecture concerning a relation between therange of $L^2$ Betti numbers and the orders of the finite subgroups of thefundamental group of the manifold.

Host: Efim Zelmanov

March 11, 2003

1:30 PM

AP&M 6438

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