119 problems

Let $n \in 2\mathbb{Z}_{\geq 1}$. Write $\Gamma^n_{\mathrm{eis}} \setminus \mathbb{R} H^n$ for the non-arithmetic hyperbolic orbifold $M({\mathscr{L}}_{\mathrm{eis}}^n,\alpha_0)$,…

Subexponential regulator conjecture.

Standard-action conjecture. Every quasi-action by $\Gamma$ on a Gromov hyperbolic metric space either has an invariant horoball or is standard.

A hyperbolic manifold is arithmetic when its fundamental group is an arithmetic lattice. A subgroup is geometrically finite if its convex core has finite volume, and a lattice is G…

Uniform stable commutator length conjecture. For each $n$ there is a constant $C(n)>0$ such that, for every arithmetic hyperbolic $n$-manifold $M$ and every non-parabolic…

Let $G$ be as above, and let $\pi=A_{\mathfrak q}$ be a cohomological representation of $G$ such that … where $\mathfrak l'_0$ is a compact Lie algebra and $\mathfrak g_1$ is a non…

Let $G$ be a semisimple algebraic group over ${\mathbb Q}$ obtained by restriction of scalars from a group of type ${}^3D_4$ or ${}^6D_4$ over a totally real number field, and supp…

Let $H$ be a connected semi-simple Lie group without almost simple factors of type $\mathrm{A}_1$ and without compact factors. For sufficiently large $x$, consider maximal irreduci…

Let $\widetilde{X}$ be a symmetric space with no Euclidean local factors, let $X=\Gamma\backslash\widetilde{X}$ be a finite-volume locally symmetric space, and let…

Grigorchuk–Lubotzky–Pyber conjecture. The two asymptotic constants coincide and equal

Let $G$ be a ${\mathord{\mathbb{Q}}}$-simple algebraic ${\mathord{\mathbb{Q}}}$-group, let $\{p_1,\ldots,p_n\}$ be a nonempty set of prime numbers, and write…

Let $G$ be a ${\mathord{\mathbb{Q}}}$-simple algebraic ${\mathord{\mathbb{Q}}}$-group, and let $G_{\mathord{\mathbb{Z}}}$ denote its integral points. A subgroup is left orderable i…

Let $N(r)$ denote the order of the outer automorphism group of the principal arithmetic subgroup $\Lambda_0^n$, in the setting of the arithmetic quotients considered in the paper.…

The torus analogue of Mazur's conjecture. The connected component of the identity of the closure of the image of $F$ in $\Gamma\backslash S({\mathbb R})$ equals the connected compo…

Fix a symmetric space $S$ of non-compact type without Euclidean de Rham factors, and let an arithmetic $S$-manifold be a quotient $M=\Gamma\backslash S$ by a torsion-free arithmeti…

Arithmetic lattice discreteness conjecture. If $W\Gamma$ is discrete, then $W\cap\Gamma$ has finite index in $W$.

Fix a nonzero rational number $q=\tfrac{s}{t}\in\mathbb{Q}\cap(-4,4)$ in lowest terms, with $\gcd(s,t)=1$. Let $\Gamma_q$ be the group generated by the root unipotents associated t…

Church–Farb–Putman conjecture. The codimension-$i$ cohomology of $\operatorname{SL}_n(\mathbb Z)$ vanishes for $n\geq i+2$; equivalently,

Let $n\geq 4$ and let $I\subseteq\{1,\dots,n\}$ satisfy the (NG)-condition. For an infinite integral domain $R$, write $\mathbf{S}^I_n(R)$ for the corresponding group, and assume t…

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…

For $\alpha\in\mathbb{C}$, let … Let $\frac{q}{p}\in\mathbb{Q}\cap(-4,4)$ with $\gcd(p,q)=1$. Write $\pi$ for the relevant reduction homomorphism. Nyberg Brodda's congruence subgro…

Let $S$ be the surface associated with a maximal volume ideal polyhedron, and let … be its holonomy representation. An arithmetic Fuchsian group is a subgroup of…

In the first case, let $\Gamma=\operatorname{SL}_2(\mathbb{Z})$ and let $p$ be any prime. In the second case, let $\Gamma$ be a cocompact arithmetic lattice of the second type in…

Modular distribution of invariants. The invariants of topo-symmetric extensions are distributed uniformly modulo $\gcd(|G|,|H|)$ in the sense above. Numerical experiments for small…

Let $W$ be a finite-rank free $\mathbb{Z}$-module, and let $\operatorname{St}(W)$ be the top reduced homology of its Tits building. For $n\geq 3$, define the integral complex with…