241 problems

Church–Farb–Putman vanishing conjecture. If $i\leq n-2$, then

Let $k$ be the base field, and let $G$ be a simply connected, almost $k$-simple, $k$-isotropic algebraic group over $k$. Let $G(k)^+$ denote the abstract subgroup of $G(k)$ generat…

Let $G$ be a simply connected group, let $k$ be a global field, let $S$ be a set of valuations, and let $\Gamma=G(\mathcal{O}_S)$. Write $\operatorname{rk}_{k_v}(G)$ for the rank o…

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

Margulis' conjecture. There exists a neighborhood $U\subset G(\R)$ of the identity such that for any irreducible arithmetic cocompact lattice $\Gamma\subset G(\R)$, the intersectio…

Thurston's conjecture. The Dehn function of $\Gamma$ is quadratic:

Let $n$ be a positive integer, let $p$ be a prime, and let $\Gamma\subset\operatorname{SL}(n,\mathbb{Z})$ be a congruence subgroup. A Hecke eigenclass is an element…

Margulis–Zimmer conjecture. Every commensurated subgroup of $\Lambda$ is either finite or $S'$-arithmetic for some $S'\subseteq S$.

Let $\ell$ be a prime number, let $K$ be a number field with $\zeta_\ell\in K$, and let $S$ be a finite set of places containing the infinite places and the places over $\ell$. The…

Serre's congruence subgroup property conjecture. Every such arithmetic lattice $\Gamma$ has the congruence subgroup property.

Let an arithmetic hyperbolic orbifold of the first type be a quotient of hyperbolic space by an arithmetic lattice of the first type. For a hyperbolic element $\gamma$, write…

Super approximation conjecture. The group $\Gamma$ has the super approximation property with respect to all positive integers if and only if $\mathbb G_0$ is perfect.

Let $k$ be a number field, let $\tilde{\mathbf{G}}$ be the ambient algebraic group, and let $\Gamma\subseteq\tilde{\mathbf{G}}(k)$ be an arithmetic subgroup. Write $V_{\infty}(k)$…

Let $G=SO(n,1)$ and let $\Gamma$ be an arithmetic lattice in $G$. Serre's conjecture. The group $\Gamma$ fails the congruence subgroup property. The notes say this conjecture was a…

Let $k$ be a global field, let $S$ be a finite set of places of $k$, and let $\mathfrak{G}$ be a simple simply connected algebraic group over $k$. The $S$-arithmetic group…

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 $n\geq 3$, let $\mathcal{O}=\mathbb{H}^{n}/\Gamma$, where $\Gamma$ is an arithmetic congruence subgroup defined by a quadratic form (or, when $n=3$, merely an arithmetic congru…

Serre's congruence subgroup property conjecture. Every higher rank group has the congruence subgroup property (CSP), meaning

Let $k$ be a number field and let $\mathbf{G}$ be a $k$-group. Let $T$ be the (possibly empty) set of finite places of $k$ at which $\mathbf{G}$ is $k_v$-anisotropic, and write … V…

Church–Farb invariant-cohomology conjecture. The $\operatorname{GL}(n,\mathbb{Z})$-invariant part of $H^*(\operatorname{IA}_n,\mathbb{Q})$ vanishes stably.

Weak Lehmer conjecture. For each $s\in\mathbb N$, there exists $\varepsilon(s)>0$ such that for any irreducible monic polynomial $P$ with integer coefficients and $s(P)\leq s$, eit…

Classification conjecture. The same classification is true for groups generated by two elements of finite order.

Let $G$ be an arithmetic hyperbolic $3$-manifold group, and consider a decreasing sequence of congruence subgroups of $G$ with trivial intersection. The first homology torsion grow…

Let $F$ be a field, let $G_1$ be an absolutely almost simple algebraic $F$-group, and let $\Gamma_1\subset G_1(F)$ be a finitely generated Zariski-dense subgroup with trace field…

Lee–Szczarba conjecture. For a prime $p$ and $n\geq 2$, the induced map