6 problems

Schmutz Schaller's conjecture. The multiplicity of the simple length spectrum is bounded above by $6$:

Commensurability conjecture. Any two hyperbolic structures with the same length spectrum are commensurable.

Let $S$ and $S'$ be hyperbolic surfaces, and let the multi-set of lengths of their simple closed geodesics be called their simple length spectrum. Simple length spectrum rigidity c…

Let $\Gamma_\mathfrak{D}$ be an arithmetic quaternion group. The complex length spectrum of $\mathbb{H}^3/\Gamma_\mathfrak{D}$ has gaps, and let $\mathcal{G}(x)$ denote the number…

Simple Hilbert length rigidity conjecture. If $\rho,\sigma\in{\mathcal H}_{d}(S)$ have the same marked simple Hilbert length spectrum, then they either agree or differ by the contr…

Let $S$ be the surface and $\rho$ a representation in $Hit_n(S)$. A representation $\rho'$ is in the Fuchsian locus of $Hit_n(S)$ when it is a Fuchsian representation viewed in…