L2-invariants

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L2-invariants play an important role in geometry and topology. In particular, they provide useful connections between (differential) geometry and questions arising from topology and even algebra, like the computation of the sign of the Euler characteristic for fundamental groups of negatively curved Kahler manifolds. The more refined of these invariants express the geometry of non-compact manifolds with topological invariants of natural compactifications, 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 the calculation of Novikov-Shubin invariants, L2-eta invariants, and L2-torsion for the natural compactifications of all locally symmetric spaces of finite volume (relating the topology of the compactification to the geometry in new ways). Moreover, we plan to extend such calculations in two directions: first to spaces which are only asymptotically locally symmetric,secondly to convex cocompact hyperbolic manifolds.