Sarnak's Density Conjecture

Let $G$ be a real, semisimple, almost-simple and simply connected Lie group, let $\Gamma_1$ be a cocompact arithmetic lattice of $G$, and let $(\Gamma_N)$ be a sequence of finite-index congruence subgroups of $\Gamma_1$, with $[\Gamma_1:\Gamma_N]\to\infty$. Let $\Pi(G)$ denote the unitary dual of $G$, and for a precompact subset $A\subset\Pi(G)$ write $M(A,\Gamma_N,p)=\sum_{\pi\in A,\,p(\pi)\ge p}m(\pi,\Gamma_N)$, where $m(\pi,\Gamma_N)$ is the multiplicity of $\pi$ in $L^2(\Gamma_N\backslash G)$. Sarnak's Density Conjecture. For every precompact subset $A\subset\Pi(G)$ and $\epsilon>0$, there exists a constant $C_{\epsilon,A}$ such that for every $N$ and $p>2$,

$$M(A,\Gamma_N,p)\le C_{A,\epsilon}[\Gamma_1:\Gamma_N]^{2/p+\epsilon}.$$

This is a density form of the Ramanujan philosophy, replacing the generally false assertion that all nontrivial representations occurring in the spectrum are tempered. The source presents it as a conjectural general framework; its resolution status is not specified here.