Constructing highly regular expanders from hyperbolic Coxeter groups

Abstract

A graph X is defined inductively to be (a0, . . . , an−1)-regular if X is a0-regular and for every vertex v of X, the sphere of radius 1 around v is an (a1, . . . , an−1)-regular graph. Such a graph X is said to be highly regular (HR) of level n if an−1 = 0. Chapman, Linial and Peled [CLP20] studied HR-graphs of level 2 and provided several methods to construct families of graphs which are expanders "globally and locally". They ask whether such HR-graphs of level 3 exist.In this paper we show how the theory of Coxeter groups, and abstract regular polytopes and their generalisations, can lead to such graphs. Given a Coxeter system (W, S) and a subset M of S, we construct highly regular quotients of the 1-skeleton of the associated Wythoffian polytope PW,M , which form an infinite family of expander graphs when (W, S) is indefinite and PW,M has finite vertex links. The regularity of the graphs in this family can be deduced from the Coxeter diagram of (W, S). The expansion stems from applying superapproximation to the congruence subgroups of the linear group W .This machinery gives a rich collection of families of HR-graphs, with various interesting properties, and in particular answers affirmatively the question asked in [CLP20].

Balanced Product Quantum Codes

“…It was proven in [25] (using results from [26]) that all hyperbolic regular tessellations {r, s} give families of sregular expander graphs. Unfortunately, we are not aware of any explicit bounds on λ 2 .…”

“…The discussion above indicates that hyperbolic expanders are related to Cayley graphs Cay PSL(2, q), { R, S, R−1 , S−1 } . This relation is made precise in [25] by defining a quasi-isometry between them.…”

Hyper-regular graphs and high dimensional expanders

“…4. Conder, Lubotzky, Schillewaert and Thilmany have made public a manuscript [5] where they (a) Independently discover the (120, 12, 5, 2)-regular family mentioned above, and calculate its expansion. (b) Manage to create an infinite families of (d 0 , .…”