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

Alexander Bendikov (1) , Christophe Pittet (2, 3) , Roman Sauer (4, 5)

1 Mathematical Institute, University of Wroclaw
2 ISCP - Interdisciplinary Scientific Center Poncelet
3 LATP - Laboratoire d'Analyse, Topologie, Probabilités
4 University of Chicago
5 UR - Universität Regensburg

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.

Mots clés: amenable group; spectral distribution; solvable group; geometric group theory; Laplace operator; exponential growth; sharp estimate

Domaines: Théorie des groupes [math.GR]; Théorie spectrale [math.SP]

Alexander Bendikov, Christophe Pittet, Roman Sauer. Spectral distribution and L2-isoperimetric profile of Laplace operators on groups. Mathematische Annalen, 2012, 354 (1), pp.43-72. ⟨10.1007/s00208-011-0724-6⟩. ⟨hal-01305024⟩