36 problems

Let $\Gamma$ be a Fuchsian group, namely a finitely generated non-elementary discrete group of isometries of the hyperbolic plane. Higman's conjecture. Every Fuchsian group surject…

Schmutz's conjecture. If the trace set of $\Gamma$ has linear growth, then $\Gamma$ is arithmetic.

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…

Sarnak's conjecture. If $\mathrm{Tr}(\Gamma)$ satisfies the bounded clustering property, then $\Gamma$ is arithmetic. If $\operatorname{Gap}(\mathrm{Tr}(\Gamma))>0$, then $\Gamma$…

Let $S=\{p_1,\ldots,p_s\}$ be a finite set of distinct primes, let $\mathbb{Z}_S$ be the subring of $\mathbb{Q}$ generated by $(p_1\cdots p_s)^{-1}$, and let … be the natural actio…

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.

Odd-genus sharpness conjecture. The lower bound is sharp for infinitely many odd genera:

Let $F_{\mathcal{A}}$ be the natural-extension map associated with a Fuchsian-group boundary map, let $A_k$ be the elliptic partition points, let $V_{k-1}$ and $V_{k+1}$ be the nei…

A Fuchsian group with cusps has a boundary map acting piecewise on ${\mathbb S}$, whose partition points determine the rectangular structure of the natural extension on geodesics.…

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 $q\in\mathbb{Q}\cap(0,4)$, and let … with $\Delta_q=\langle A,Q_q\rangle$. Kim–Koberda's non-freeness conjecture. The group $\Delta_q$ is not free for every…

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 ${\,\overline{\!A\,}\,}$ be a multi-parameter with $A_k\in[P_k,Q_k]$. Let $F_{\,\overline{\!A\,}\,}$ be the associated map, let $\Omega_{\,\overline{\!A\,}\,}$ be its conjectur…

Let ${\,\overline{\!A\!}\,}$ be a multi-parameter with $A_k\in[P_k,Q_k]$, and let $F_{\,\overline{\!A\,}\,}$ be the associated map on ${\mathbb S}\times{\mathbb S}\setminus\Delta$.…

Let $g$ be the genus, let ${\mathcal F}\in {\mathcal T}(g)$ be a fundamental polygon, and let ${\,\overline{\!A\!}\,}$ be a multi-parameter with $A_k\in[P_k,Q_k]$. Let…

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…

Corner-point conjecture. The corner points of $\Omega_{\overline{A}}$ for extremal $\overline{A}$ should be of the form $(\cdot,Q_i)$ for lower corners and $(\cdot,P_i)$ for upper…

Let $\Gamma_0$ be a cocompact Fuchsian group, let $\Gamma\vartriangleleft\Gamma_0$, and let $G=\Gamma_0/\Gamma$ be a hyperbolic group. Let $\delta(\Gamma)$ be the critical exponent…

Let $G$ be a locally discrete subgroup of $\mathsf{Diff}_+^\omega(\mathbb{S}^1)$, where $\mathsf{Diff}_+^\omega(\mathbb{S}^1)$ denotes the group of orientation-preserving real-anal…