L2-invariants of groups

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L2-invariatits play an important role in geometry and topology. In particular, they provide useful connections between questions arising in the area of topology, algebra and (differential) geometry. For example, one can use L2 -invariants to prove cases of the Kaplansky zero-divisor conjecture for group rings of torsionfree groups or estimate the clericiency of a discrete group. Generalizations and relinements of these invariants have been introduced for quantum groups, via center valued versions, or in terms of Lp-cohomology, together with the corresponding homological algebra. The more refined of these invariants allow to express the geometry of noncompact manifolds in terms of topological invariants of natural conipactilications, via the explicit calculation of L2-invariants. It turns out, however, that except for L2-Betti numbers there is a lack of explicitly calculated examples. Our goals in this project are: line study of the center valued L2-Betti numbers and their possible values; statement (and proof in special cases) of a corresponding Atiyah conjecture; relation of this to the ring theoretic properties of the division closure of the group ring detailed investigation of L2-invariants of quantum groups, including the study of the algebraic and arithmetic properties of quantum groups the calculation of Novikov-Shubin invariants, L2-eta invariants, and L2-torsion for the natural compactifications of locally symmetric spaces of finite volume (relating the topology of the compactification to the geometry in new ways).