We prove the K-and the L-theoretic Farrell-Jones conjecture with coefficients in additive categories and with finite wreath products for arbitrary lattices in virtually connected Lie groups.

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We study the existence of lattices in a higher rank simple Lie group which are profinitely but not abstractly commensurable. We show that no such examples exist for the complex forms of type E 8 , F 4 , and G 2 . In contrast, there are arbitrarily many such examples for all other higher rank Lie groups, except possibly SL 2n+1 (R), SL 2n+1 (C), SL n (H), and any form of type E 6 .

We compute L 2-invariants of certain nonuniform lattices in semisimple Lie groups by means of the Borel-Serre compactification of arithmetically defined locally symmetric spaces. The main results give new estimates for Novikov-Shubin numbers and vanishing L 2-torsion for lattices in groups with even deficiency. We discuss applications to Gromov's zero-in-the-spectrum conjecture as well as to a proportionality conjecture for the L 2-torsion of measure-equivalent groups.

We assign real numbers to finite sheeted coverings of compact CW complexes designed as finite counterparts to the Novikov-Shubin numbers. We prove an approximation theorem in the case of virtually cyclic fundamental groups employing methods from Diophantine approximation.(2) AA * : (ℓ 2 G) r → (ℓ 2 G) r given by x → xAA * . Here the matrix A * is obtained from A by transposing and applying the canonical involution ( λ g g) * = λ g g −1 to the entries. Let {E AA * λ } λ≥0 be the family of equivariant spectral projections obtained from r (2) AA * by Borel functional calculus, E AA * λ = χ [0,λ] (r (2) AA * ), where χ [0,λ] is the