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.