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In their seminal work from the 80’s,ÌýLubotzky, Phillips andÌýSarnakÌýgave explicit constructions of topological generators for PU(2) with optimal covering properties. In this talk I will describe some recent works that extend the construction of LPS to higher rank compact Lie groups.Ìý

A key ingredient in the work of LPS is the Ramanujan conjecture for U(2), which follows fromÌýDeligne'sÌýproof of the Ramanujan-Petersson conjecture forÌýGL(2). Unfortunately, the naive generalization of theÌýRamanujan conjecture is false for higher rank groups. Following a program initiated byÌýSarnakÌýin the 90's, we prove a densityÌýhypothesis and use it as a replacementÌýof the naive Ramanujan conjecture.

This talk is based on some joint works withÌýOriÌýParzanchevski and Amitay Kamber.