Geodesic
$L^2$-invariants of locally symmetric spaces
Documenta mathematica, Tome 7 (2002), pp. 219-237
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Let X=G/K be a Riemannian symmetric space of the noncompact type, Γ⊂G a discrete, torsion-free, cocompact subgroup, and let Y=Γ\X be the corresponding locally symmetric space. In this paper we explain how the Harish-Chandra Plancherel Theorem for L2(G) and results on (g,K)-cohomology can be used in order to compute the L2-Betti numbers, the Novikov–Shubin invariants, and the L2-torsion of Y 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 m=rkC​G−rkC​K of G. In particular, we show the nonvanishing of the L2-torsion of Y whenever m=1.
DOI : 10.4171/dm/125
Classification : 22E46, 57R19, 58J35
Mots-clés : locally symmetric spaces, L2-cohomology, Novikov-shubin invariants, L2-torsion, relative Lie algebra cohomology 
@article{10_4171_dm_125,
     author = {Martin Olbrich},
     title = {$L^2$-invariants of locally symmetric spaces},
     journal = {Documenta mathematica},
     pages = {219--237},
     year = {2002},
     volume = {7},
     doi = {10.4171/dm/125},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/125/}
}
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Martin Olbrich. $L^2$-invariants of locally symmetric spaces. Documenta mathematica, Tome 7 (2002), pp. 219-237. doi: 10.4171/dm/125

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