Spectral distribution and $L^2$-isoperimetric profile of Laplace operators on groups

TL;DR

This paper establishes a formula linking the spectral distribution of Laplace operators to the $L^2$-isoperimetric profile of groups, enabling geometric group theory techniques to estimate spectral properties and providing sharp results for solvable groups.

Contribution

It introduces a new formula connecting spectral distribution and isoperimetric profile, extending estimates to solvable groups and proving invariance under measure changes.

Findings

Derived a formula relating spectral distribution to isoperimetric profile.

Provided sharp spectral distribution estimates for solvable groups.

Proved invariance of spectral distribution under measure changes with finite second moment.

Abstract

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.