Expansion in finite simple groups of Lie type

Abstract

Two short seminal papers of Margulis used Kazhdan's property (T ) to give, on the one hand, explicit constructions of expander graphs, and to prove, on the other hand, the uniqueness of some invariant means on compact simple Lie groups. These papers opened a rich line of research on expansion and spectral gap phenomena in finite and compact simple groups. In this paper we survey the history of this area and point out a number of problems which are still open.

On a Cheeger type inequality in Cayley graphs of finite groups

“…The strategy of the proof closely follows the combinatorial arguments of Breuillard-Green-Guralnick-Tao in [BGGT15].…”

“…The Cayley graph is bipartite if and only if there exists an index two subgroup H of G which is disjoint from S. See Proposition 2.6. It was observed in [BGGT15](Appendix E) that if C(G, S) is an expander graph and is non-bipartite, then the spectrum of T is not only bounded away from 1 but also from −1. Here we show that Theorem 1.4.…”

Hypergraph expanders from Cayley graphs

“…is Ω(|S| |Ii|−1 ). 3 Furthermore, Remark 2.5 implies that we can still find this many walks with no coordinate u i equal to u Ij for j > i.…”

Aldous’s spectral gap conjecture for normal sets

“…This question is asked for all finite non abelian simple groups in [Lub12, Problem 2.28] (with uniform ε for all), and is proven in [BGGT15] for finite simple group of Lie type and bounded Lie rank. However, the case of A n remains wide open.…”