L^2-invariants of locally symmetric spaces

Abstract

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.

arXiv comment18 pages; The introduction now contains more precise references to the work of other people on Novikov-Shubin invariants, in particular to a recent paper of Lohoue and Mehdi

Cited by in corpus (6)

• Spectral density and Sobolev inequalities for pure and mixed states
• Lp-cohomology of symmetric spaces
• L2-invariants of nonuniform lattices in semisimple Lie groups
• Analytic torsion and L^2-torsion of compact locally symmetric manifolds
• Weak notions of normality and vanishing up to rank in L2-cohomology
• Asymptotics of automorphic spectra and the trace formula