  Main theorems of the article concern the problem of M. Atiyah on possible
values of l^2-Betti numbers. It is shown that all non-negative real numbers are
l^2-Betti numbers, and that "many" (for example all non-negative algebraic)
real numbers are l^2-Betti numbers of simply connected manifolds with respect
to a free cocompact action. Also an explicit example is constructed which leads
to a simply connected manifold with a transcendental l^2-Betti number with
respect to an action of the threefold direct product of the lamplighter group
Z/2 wr Z. The main new idea is embedding Turing machines into integral group
rings. The main tool developed generalizes known techniques of spectral
computations for certain random walk operators to arbitrary operators in
groupoid rings of discrete measured groupoids.
