14 problems

A closed hyperbolic manifold is a quotient of hyperbolic space by a cocompact lattice, and its systole is the length of its shortest closed geodesic. An arithmetic hyperbolic manif…

Let $\mathcal{M}_g$ be the moduli space of genus-$g$ hyperbolic surfaces. A set of geodesics on a hyperbolic surface is filling if the geodesics cut the surface into disks, and a s…

Let $S_k$ be the compact arithmetic hyperbolic surfaces described in the source, and let the systolic inequality compare the systole of a hyperbolic surface with its genus or area.…

Let ${\mathcal T}_g$ be the Teichmüller space of closed hyperbolic surfaces of genus $g$, and let ${\mathcal X}_g$ be the subset of surfaces whose systoles fill. For every…

Let $M$ be an arithmetic hyperbolic $2$- or $3$-manifold, and let its systole be the length of its shortest non-contractible closed geodesic. Short Geodesics Conjecture. There is a…

Let $\Sigma$ be a compact hyperbolic surface, and let $I_k(\Sigma)$ denote the maximal self-intersection number of a $k$-systole on $\Sigma$. Erlandsson–Parlier's conjecture. One h…

Extremal length systole maximum conjecture. The function

Aougab–Huang's filling pair systole conjecture. For every $M_g\in\mathcal{M}_g$,

Margulis's conjecture. There exists a uniform lower bound for the systoles of arithmetic locally symmetric manifolds of finite volume.

Dimension conjecture. For every $\varepsilon>0$, there exists an integer $g\geq 2$ such that $\mathcal{X}_g$ has dimension at least $(6-\varepsilon)g$.

Let $S_g$ be a closed orientable surface of genus $g$, let $\mathcal{M}_g$ denote its moduli space of hyperbolic metrics, and let $m_g$ be the minimum perimeter of a regular, right…

Large-collar failure conjecture. A non-separating systole of a non-compact maximal surface can fail to have a large collar.

Let $\operatorname{sys}(g,n)$ denote the maximal systole among finite-area hyperbolic surfaces of genus $g$ with $n$ cusps, and let $(g_p)$ be the sequence of genera associated wit…

Let $\mathcal M_g$ denote the moduli space of closed hyperbolic surfaces of genus $g$, and let $\operatorname{Kiss}(S)$ be the number of systoles of $S$. Schmutz Schaller's kissing…