LERF and the Lubotzky-Sarnak Conjecture

Producing the graphs of Lubotzky Phillips and Sarnak in Matlab

This document describes the details of how the X graphs of Lubotzky, Phillips, and Sarnak are produced in Matlab. Included is a (hopefully minimal) description of the theory behind the construction. The graphs were originally defined only when p and q are both congruent to 1 mod 4, but an extension of the construction to all odd primes is given in the book by Davidoff, Sarnak, and Valette, in w...

The Ramanujan Property for Regular Cubical Complexes

We consider cubical complexes which are uniformized by an ordered product of regular trees. For these we define the notion of being Ramanujan, generalizing the one-dimensional definition introduced by Lubotzky, Phillips, and Sarnak [15]. As in [12], we also allow local systems. We discuss the significance of this property, and then we construct explicit arithmetic examples using quaternion alge...

Ramanujan graphs

Networks via Hilbert Modular Forms

Ramanujan graphs, introduced by Lubotzky, Phillips and Sarnak, allow the design of efficient communication networks. In joint work with B. Jordan we gave a higher-dimensional generalization. Here we explain how one could use this generalization to construct efficient communication networks which allow for a number of verification protocols and for the distribution of information along several c...

Generalized Triangle Groups, Expanders, and a Problem of Agol and Wise

Answering a question asked by Agol and Wise, we show that a desired stronger form of Wise’s malnormal special quotient theorem does not hold. The counterexamples are generalizations of triangle groups, built using the Ramanujan graphs constructed by Lubotzky–Phillips–Sarnak.