On Sarnak’s Density Conjecture and Its Applications

Abstract Sarnak’s density conjecture is an explicit bound on the multiplicities of nontempered representations in a sequence cocompact congruence arithmetic lattices semisimple Lie group, which motivated by work Sarnak and Xue ([58]). The goal this to discuss similar hypotheses, their interrelation applications. We mainly focus two properties – spectral spherical hypothesis geometric Weak injective radius property. Our results are strongest p -adic case, where we show that equivalent, both imply general hypothesis. One possible application either or property optimal lifting ([57]). Conjecturally, all those should hold great generality. hope will motivate proofs new cases.