8 problems

Schmutz's conjecture. If $\mathrm{Tr}(\Gamma)$ has linear growth, then $\Gamma$ is arithmetic.

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 $G(\mathbb{A}_F)$.…

Simpson's conjecture. Every such representation has integral image. This connects rigidity of local systems on projective varieties with arithmeticity; the source presents it as Si…

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…

Clozel's conjecture. The following are equivalent:

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