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On the differential form spectrum of hyperbolic manifolds. (English) Zbl 1170.53309

Summary: We give a lower bound for the bottom of the \(L^2\) differential form spectrum on hyperbolic manifolds, generalizing thus a well-known result due to Sullivan and Corlette in the function case. Our method is based on the study of the resolvent associated with the Hodge-de Rham Laplacian and leads to applications for the (co)homology and topology of certain classes of hyperbolic manifolds.

MSC:

53C35 Differential geometry of symmetric spaces
22E40 Discrete subgroups of Lie groups
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
57T15 Homology and cohomology of homogeneous spaces of Lie groups
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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