4 problems

Schmutz's conjecture. If the trace set of $\Gamma$ 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 $\Gamma$…

Sarnak's conjecture. Bounded clustering characterizes arithmetic Fuchsian groups, and uniform positive separation of traces forces a group to be derived from a quaternion algebra.

Let $\G$ be a subarithmetic hyperbolic group with growth exponent $\ga$, and let $\cO$ be the ring of integers of its trace field. The group $\G$ asymptotically length-saturates if…