Abstract

The symmetric space of $SL_2®$ is the hyperbolic plane, and the fact that $SL_2(Z)$ is a lattice in $SL_2®$ implies that after taking a quotient we get a finite volume hyperbolic surface.

When $SL_2®$ is replaced by the p-adic group $SL_2(Q_p)$ the symmetric space is a (q+1)-regular Bruhat-Tits tree. Ihara, Margulis and Lubotzky-Phillips-Sarnak observed that when $SL_2(Z)$ is replaced by a lattice coming from a quaternion algebra one gets a (q+1)-regular graph. Using deep number theoretic results from the theory of automorphic forms, related to the classical Ramanujan conjecture, they showed that the resulting graphs are expanders with an optimal spectral gap, i.e., “Ramanujan graphs”.

The number-theoretic Ramanujan graphs have a lot of combinatorial applications and were generalized to various combinatorial number theoretic constructions.

However, there are some notorious open questions about those constructions, such as the vertex expansion of number theoretic Ramanujan graphs.

In the talk, I will describe how one can construct extermal substructures of some number-theoretic structures, which provides counterexamples for many open problems. The idea is group-theoretic and simple – we use closed orbits of subgroups, when those subgroups are available. The implementation of the idea requires some number theory.

I will try to appeal to a wide audience, and focus on the group theory involved.

Based on joint work with Tali Kaufman.