  Sarnak's Density Conjecture is an explicit bound on the multiplicities of
non-tempered representations in a sequence of cocompact congruence arithmetic
lattices in a semisimple Lie group, which is motivated by the work of Sarnak
and Xue. The goal of this work is to discuss similar hypotheses, their
interrelation and applications. We mainly focus on two properties - the
spectral Spherical Density Hypothesis and the geometric Weak Injective Radius
Property. Our results are strongest in the p-adic case, where we show that the
two properties are equivalent, and both imply Sarnak's General Density
Hypothesis. One possible application is that either the Spherical Density
Hypothesis or the Weak Injective Radius Property imply Sarnak's Optimal Lifting
Property. Conjecturally, all those properties should hold in great generality.
We hope that this work will motivate their proofs in new cases.
