$L^2$-invariants and geometric group theory, I

We give a short introduction to $L^2$-invariants and then
discuss their applications to geometric group theory. In the first 45
minutes we will explain and discuss recent results and in the second part we could 

talk on request about proofs, outstanding conjectures and future  projects. The
results we want to discuss consider the relationship of  the first
$L^2$-Betti number to deficiency, the construction of  infinite finitely
generated non-amenable residually finite torsion  groups, the growth of
homology in towers of coverings and the clash of  the Heegard Genus
Conjecture about 3-manifolds and the Fixed Prize  Conjecture about orbit
equivalence relations.