19 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 $\Gamma$…

Representation-growth conjecture. (a) There exists a $2A$-periodic sequence $c_{q,1},c_{q,2},\ldots$ of positive numbers such that

Let $\mu$ be a countably supported measure whose support generates a Fuchsian group $\Gamma \subset PSU(1,1)$ of second type. Let a solution be a function satisfying the equation r…

Let $\Gamma$ be a Fuchsian group, let $\Sigma$ be a double coset of $\Gamma$, and let $X$ be a $\Gamma$-conjugacy class contained in $\Sigma$. For $M\in X$, define the discriminant…

Sarnak's conjecture. If $\mathrm{Tr}(\Gamma)$ satisfies the B-C property, then $\Gamma$ is arithmetic. If $\operatorname{Gap}(\mathrm{Tr}(\Gamma))>0$, then $\Gamma$ is derived from…

Let $G$ be the group of isometries of the hyperbolic plane $\mathbb{H}^2$. Let $\mu$ be a finitely supported probability on $G$ whose support generates, as a semigroup, a discrete…

Let $(X_n)$ be a finite-range admissible random walk generated by a probability measure $\mu$ on a cocompact Fuchsian group $\Gamma$, and let $\nu_\mu$ be its hitting measure on th…

Let $\Gamma$ be a Fuchsian group of the first kind with cusps, and let $\phi$ be a Maaß cusp form for $\Gamma$ of integer weight. For $\Gamma=\operatorname{SL}(2,\mathbb{Z})$, the…

Let $\Gamma$ be a discrete subgroup of $\mathrm{PSL}_2(\mathbb{R})$ whose fundamental domain has finitely many sides; such a group is geometrically finite. Geometrically finite geo…

Let $\Gamma$ be a subgroup of $\mathrm{PSL}_2(\mathbb{Z})$ of finite index, and let its geodesic-covering number be defined using a fundamental domain. Finite-index modular subgrou…

Algorithmic termination conjecture. For all $s,r\in\mathbb{N}$ satisfying $s/r<4$, there exists $M>0$ such that

Relation-number conjecture. Every rational number in $(-4,4)$ is an $\ell$-step relation number for some $\ell\geq 0$.

Let $\Gamma$ be a Fuchsian group, and let $G$ be a finite simple classical group. A randomly chosen homomorphism in ${\rm Hom}(\Gamma,G)$ is an epimorphism with a probability that…

Let $G\leq \operatorname{Diff}_+^\omega(\mathbf{S}^1)$ be a finitely generated locally discrete group. A locally discrete group's structure conjecture predicts both that the action…

Let $G$ be a Fuchsian group of the first kind, and let $G^\phi$ be a b-group arising from a point $\phi$ in the relevant deformation space. A group $G^\phi$ is called a boundary gr…

Let $\Gamma<\operatorname{PSL}_2(\mathbb{R})$ be a lattice, and let $m$ be a finitely supported measure whose support generates $\Gamma$. Let the corresponding stationary measure b…

Let $\overline{C}$ be a smooth projective curve of genus $0$ defined over a number field, and let $({\mathcal E}, \nabla)$ be a flat vector bundle of rank $2$ which is globally nil…

Let $\overline{C}$ be a smooth projective curve defined over a number field. Let $({\mathcal E}, \nabla)$ be an indigenous bundle on $\overline{C}$, and let…