Combinatorics via closed orbits: number theoretic Ramanujan graphs are not unique neighbor expanders

The question of finding expander graphs with strong vertex expansion properties such as unique neighbor expansion and lossless expansion is central to computer science. A barrier to constructing these is that strong notions of expansion could not be proven via the spectral expansion paradigm. A very symmetric and structured family of optimal spectral expanders (i.e., Ramanujan graphs) was constructed using number theory by Lubotzky, Phillips and Sarnak, and and was subsequently generalized by others. We call such graphs Number Theoretic Ramanujan Graphs. These graphs are not only spectrally optimal, but also posses strong symmetries and rich structure. Thus, it has been widely conjectured that number theoretic Ramanujan graphs are lossless expanders, or at least unique neighbor expanders. In this work we disprove this conjecture, by showing that there are number theoretic Ramanujan graphs that are not even unique neighbor expanders. This is done by introducing a new combinatorial paradigm that we term the closed orbit method. The closed orbit method allows one to construct finite combinatorial objects with extermal substructures. This is done by observing that there exist infinite combinatorial structures with extermal substructures, coming from an action of a subgroup of the automorphism group of the structure. The crux of our idea is a systematic way to construct a finite quotient of the infinite structure containing a simple shadow of the infinite substructure, which maintains its extermal combinatorial property. Other applications of the method are to the edge expansion of number theoretic Ramanujan graphs and vertex expansion of Ramanujan complexes. Finally, in the field of graph quantum ergodicity we produce number theoretic Ramanujan graphs with an eigenfunction of small support that corresponds to the zero eigenvalue. This again contradicts common expectations. The closed orbit method is based on the well-established idea from dynamics and number theory of studying closed orbits of subgroups. The novelty of this work is in exploiting this idea to combinatorial questions, and we hope that it will have other applications in the future.