- Spectral distribution and L2-isoperimetric profile of Laplace operators on groups doi link

Auteur(s): Bendikov Alexander, Pittet Christophe, Sauer Roman

(Article) Publié: Mathematische Annalen, vol. 354 p.43-72 (2012)

Ref HAL: hal-01305024_v1
Ref Arxiv: 0901.0271
DOI: 10.1007/s00208-011-0724-6
Ref. & Cit.: NASA ADS
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We give a formula relating the $L^2$-isoperimetric profile to the spectral distribution of the Laplace operator associated to a finitely generated group $\Gamma$ or a Riemannian manifold with a cocompact, isometric $\Gamma$-action. As a consequence, we can apply techniques from geometric group theory to estimate the spectral distribution of the Laplace operator in terms of the growth and the F{\o}lner's function of the group, generalizing previous estimates by Gromov and Shubin. This leads, in particular, to sharp estimates of the spectral distributions for several classes of solvable groups. Furthermore, we prove the asymptotic invariance of the spectral distribution under changes of measures with finite second moment.

Commentaires: 22 pages; changed title; improved exposition and gave more details in some of the proofs.To the memory of Andrzej Hulanicki.