9 problems

Kapovich's discreteness conjecture. The set

Let $S$ be a surface, and let $\operatorname{Convex}(S)$ denote the space of convex curve functionals on $S$, equipped with the product topology inherited from its values on the cu…

Let $S$ be a surface with cusps and geodesic boundary. Let $\mathcal{C}_{\mathrm{cusp}}(S)$ be the class of curves and arcs in the cusped setting, and let…

Let $S$ be a surface with cusps and geodesic boundary. Write $\operatorname{Curr}_{\mathrm{cusp}}(S)$ and $\operatorname{Curr}_{\mathrm{open}}(S)$ for the corresponding spaces of g…

Reid's conjecture. $$ -Finsler metrics are determined by their length spectra, and these length spectra form a dense subset of

Let $G$ be a hyperbolic group, let $\partial G$ be its hyperbolic boundary, and let $\partial^2G=\partial G\times\partial G\setminus\operatorname{diag}$. A geodesic current is a…

Let $\Gamma$ be a hyperbolic group acting isometrically on an $\mathbb{R}$-tree. The associated stable translation length function is the function assigning to each element its sta…

Let $\Sigma$ be a hyperbolic surface, let $\mathcal{L}_{\Sigma}$ be its Liouville current, and let $\operatorname{EL}_{\Sigma}$ denote the continuous extension of extremal length t…

Let $S$ be the underlying surface and let $\mathrm{Flat}(S)$ denote the space of flat metrics on $S$. For $\varphi\in\mathrm{Flat}(S)$, let $L_{\varphi}$ be its Liouville geodesic…