8 problems

Sarnak's conjecture. If $\mathrm{Tr}(\Gamma)$ satisfies the bounded clustering property, then $\Gamma$ is arithmetic. If $\operatorname{Gap}(\mathrm{Tr}(\Gamma))>0$, then $\Gamma$…

Let $G$ be a connected reductive algebraic group defined over a number field $F$, and let $\pi=\bigotimes_v\pi_v$ be an irreducible automorphic representation of $G(\mathbb{A}_F)$.…

Let $X$ be a smooth algebraic variety, and let $\mathcal F$ be a semisimple perverse sheaf on $X$. An absolute $\overline{\mathbb Q}$-point is a complex perverse sheaf satisfying t…

Let $G$ be a simple non-compact real algebraic group and let $H<G$ be a proper simple non-compact subgroup. Consider the question of finding sufficient conditions ensuring that eve…

For every $n\ge 2$, consider arithmetic lattices $\Gamma_1,\Gamma_2<\operatorname{SU}(n-1,1)$ and $\Gamma_3<\operatorname{SU}(n-2,1)$. Suppose $\Gamma_3$ is isomorphic to subgroups…

Let $n$ be a positive integer. Nonarithmetic lattice existence conjecture. For each $n$, the group $\operatorname{PU}(n,1)$ contains a nonarithmetic lattice. Nonarithmetic lattices…

Lafont–McReynolds' conjecture. Having the stronger progression property characterizes the arithmeticity of $M$.

The $C$-arithmeticity conjecture. The representation $cpi$ is $C$-arithmetic if and only if it is $C$-algebraic.