Universal inequalities for Dirichlet eigenvalues on discrete groups

TL;DR

This paper establishes universal inequalities for Dirichlet eigenvalues of Laplacians on subsets of discrete groups, extending classical results from Riemannian geometry to graph structures like Cayley graphs and regular trees.

Contribution

It introduces Yang-type universal inequalities for Laplacian eigenvalues on Cayley graphs of finitely generated amenable groups and regular trees, bridging geometric analysis and discrete group theory.

Findings

Proved Yang-type inequalities for Cayley graphs.

Extended universal eigenvalue inequalities to regular trees.

Connected spectral properties of discrete groups with classical geometric results.

Abstract

We prove universal inequalities for Laplacian eigenvalues with Dirichlet boundary conditions on subsets of certain discrete groups. The study of universal inequalities on Riemannian manifolds was initiated by Weyl, Polya, Yau, and others. Here we focus on a version by Cheng and Yang. Specifically, we prove Yang-type universal inequalities for Cayley graphs of finitely generated amenable groups, as well as for the d-regular tree (simple random walk on the free group).