  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).
