Let be a Riemannian symmetric space of the noncompact type, a discrete, torsion-free, cocompact subgroup, and let be the corresponding locally symmetric space. In this paper we explain how the Harish-Chandra Plancherel Theorem for and results on -cohomology can be used in order to compute the -Betti numbers, the Novikov–Shubin invariants, and the -torsion of in a uniform way thus completing results previously obtained by Borel, Lott, Mathai, Hess and Schick, Lohoue and Mehdi. It turns out that the behaviour of these invariants is essentially determined by the fundamental rank of . In particular, we show the nonvanishing of the -torsion of whenever .
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Martin Olbrich, -invariants of locally symmetric spaces. Doc. Math. 7 (2002), pp. 219–237DOI 10.4171/DM/125