18 problems

Commensurator discreteness conjecture. The group $\operatorname{Comm}_{\mathrm{PSL}_2(\mathbb{R})}(\Phi(H))$ is discrete provided that $H\neq\Phi(H)$. More generally,

Let $n$ be an odd integer with $n\geqslant 5$. Let $a_n$ and $b_n$ denote the cardinalities of the two discrete orbits of primitive $n$-square-tiled surfaces in ${\mathcal{H}}(2)$,…

Let $H^+$ be the upper half-plane, and define … The groups $GL_2^+(\mathbb Q)$ and $B^+(\mathbb Q)$ act on $H^+$ by fractional linear transformations. Winkelmann's upper-half-plane…

Let $\Gamma=\operatorname{PSL}_2(\mathbb{Z})$, let $\Gamma^\prime\subset\Gamma$ be a finite-index subgroup corresponding to a finite cover $\mathbf{M}^\prime\to\mathbf{M}$, and let…

Let $p$ be a prime and let $n\geq 3$. A Farey sequence of $\Gamma_0(p^n)$ is denoted by $\{e_i\}$. Optimal upper-bound conjecture. The group $\Gamma_0(p^n)$ has a Farey sequence…

Sharpness conjecture. The number of such solutions is at most

Let $\alpha,\beta\in\mathbb{R}\setminus\mathbb{Q}$ be irrational. Say that they are eventually $\operatorname{PGL}_2(\mathbb{Z})$-equivalent if there exists $m\in\mathbb{N}$ such t…

For $N\geq 2$, let $t$ be the number of positive divisors of $N$, let $g_1,\ldots,g_r$ be the generators specified in the surrounding setup, and let $\sigma_{N,l}$ be the associate…

Let $N\geq 2$ be an integer, and write $\beta(N)=\beta(N,N)$. Formula conjecture. 1. If $N\not\equiv 1,9\pmod{24}$, then $\beta(N)$ is periodic in $N$ with period $24$, with values…

Let $N\geq 2$ and $1<l\mid N$. The homomorphism $\sigma_{N,l}$ has image $\operatorname{im}(\sigma_{N,l})=\beta(N,l)\mathbb{Z}$ for a unique positive integer $\beta(N,l)$. Dependen…

Noncongruence-surjection conjecture. If $G$ is a nonabelian finite simple group, then there exists a surjection $\varphi:F_2\to G$ such that $\Gamma_\varphi$ is noncongruence. Equi…

Let $\mathcal{O}$ be an origami of degree $d$ and genus $2$, let $\Gamma\subseteq\operatorname{SL}_2(\mathbb{Z})$ be its Kontsevich–Zorich monodromy, and let $\mathcal{B}_d$ denote…

Let $c_1,c_2,\ldots,c_k$ be integers with $c_i\geq 2$, and let $M_q(c_1,c_2,\ldots,c_k)$ be the associated matrix. Two sequences are considered equivalent when one is obtained from…

Let $c_1,c_2,\ldots,c_k$ be integers, and let $M_q(c_1,c_2,\ldots,c_k)$ be the associated matrix in $\mathrm{PSL}_q(2,\mathbb{Z})$. Leclere–Morier-Genoud trace unimodality conjectu…

Let $L$ be the ambient subgroup, let $h(t)$ count the $\operatorname{SL}_2(\mathbb{Z})$-conjugacy classes of hyperbolic matrices in $L$ with trace $t$, and let $\ell([g])$ denote t…

Let $\Gamma(2)'$ be the commutator subgroup of $\Gamma(2)$, and let $\operatorname{Tr}(\Gamma(2)')$ denote its trace set. The commutator width of an element is the minimum number o…

Let $L$ be the ambient subgroup under consideration, and let $h(t)$ be the number of $\operatorname{SL}_2(\mathbb{Z})$-conjugacy classes of hyperbolic matrices in $L$ with trace…

Let $\Gamma(2)'$ be the commutator subgroup of the level-2 principal congruence subgroup $\Gamma(2)\subset\operatorname{SL}_2(\mathbb{Z})$, and let an integer $t$ be admissible if,…