The Ramanujan Property for Simplicial Complexes

Understand

Let $G$ be a topological group acting on a simplicial complex $\mathcal{X}$ satisfying some mild assumptions.

• For example, consider a $k$-regular tree and its automorphism group, or more generally, a regular affine Bruhat-Tits building and its automorphism group.
• We define and study various types of high-dimensional spectra of quotients of $\mathcal{X}$ by subgroups of $G$.
• These spectra include the spectrum of many natural operators associated with the quotients, e.g.