L^2-invariants of locally symmetric spaces

Details L^2-invariants of locally symmetric spaces L^2-invariants of locally symmetric spaces

2000-09-04

arxiv.org/abs/math/0009039v2

Mathematics

Differential Geometry

18 pages; The introduction now contains more precise references to the work of other people on Novikov-Shubin invariants, in p

Scientific paper

We explain how the Harish-Chandra Plancherel Theorem and results in relative Lie algebra cohomology can be used in order to compute in a uniform way the $L^2$-Betti numbers, the Novikov-Shubin invariants, and the $L^2$-torsion of compact locally symmetric spaces thus completing results previously obtained by Borel, Lott, Mathai, Hess and Schick. It turns out that the behaviour of these invariants is essentially determined by the fundamental rank of the group of isometries of the corresponding globally symmetric space. In particular, we show the nonvanishing of the $L^2$-torsion whenever the fundamental rank is equal to 1.

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