11:10 Uriya First, University of British Columbia
Ramanujan Complexes
In 1988, Lubotzky, Philips, Sarnak and independently Margulis
constructed optimal expanding graphs, i.e. Ramanujan graphs. Beside
being supreme expanders, these graphs enjoy many "good" combinatorial
properties such as large girth and large chromatic number. The
construction was generalized in 2005 to simplicial complexes by
Lubotzky, Samuels and Vishne, to produce the so-called Ramanujan
complexes. Both constructions were based on translating the spectral
properties of the graph or simplicial complex into properties of a
certain representation of a certain group (PGL_d(F) for a local field
F) acting on the universal cover of the complex.
I will discuss recent generalizations of the aforementioned works: Given a simplicial complex X and a topological group G acting on X. One can associate various types of spectra with quotients of X by subgroups of G, and define Ramanujan quotients of X accordingly. The Ramanujan property can be rephrased as a representation-theoretic condition on a certain unitary representation of G. Applying this together with deep results about automorphic representations, we obtain new examples of Ramanujan complexes (among them are Ramanujan graphs which are apparently new). We further show that Ramanujan complexes of Lubotzky, Samuels and Vishne are Ramanujan with respect to high-dimensional operators, e.g. the high-dimensional Laplacians.