群上Laplace算子的谱分布和$L^2$ -等周轮廓
Spectral distribution and $L^2$-isoperimetric profile of Laplace operators on groups
摘要：
本文给出了一个公式，它将$L^2$ -等周轮廓与Laplace算子的谱分布联系起来，该算子与一个有限生成群$Γ$或一个具有余紧等距$Γ$ -作用的黎曼流形有关.因此，我们可以应用几何群论中的技术，根据群的增长和Følner函数来估计拉普拉斯算子的谱分布，推广了Gromov和Shubin以前的估计。这导致，特别是尖锐的估计的谱分布的几类可解群。此外，我们证明了有限二阶矩测度变化下的谱分布的渐近不变性。
英文摘要：
We give a formula relating the $L^2$-isoperimetric profile to the spectral distribution of the Laplace operator associated to a finitely generated group $Γ$ or a Riemannian manifold with a cocompact, isometric $Γ$-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 Følner'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.
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