49 problems

Let $G={\rm SL}(n,\mathbb{R})$, $\Gamma={\rm SL}(n,\mathbb{Z})$, and $X=G/\Gamma$ for $n\geq 3$. Let $A<G$ be the group of positive diagonal matrices with determinant one. A Borel…

Let $G$ be the product of $n\geq 2$ copies of $\operatorname{SL}(2,\mathbb{R})$. Let $U_1$ and $U_2$ be the products of, respectively, upper- and lower-unipotent one-parameter subg…

Fix $\rho > 0$. Let $G=\mathbf{G}(\mathbb{R})$ be an $\mathbb{R}$-split real algebraic group, let $\Gamma < G$ be an arithmetic lattice, and let $H$ be a maximal $\mathbb{R}$-split…

Let $n \geq 3$ and let $H$ be the diagonal subgroup acting on $\operatorname{PGL}_n(\mathbb{Z}) \backslash \operatorname{PGL}_n(\mathbb{R})$. Finiteness conjecture for periodic orb…

Let $n \geq 3$, let $G \cong \operatorname{PGL}_n(\mathbb{R})$, and let $\Gamma < G$ be a lattice as in examples (L-1) or (L-2) of the source. Let $H$ be the relevant diagonal subg…

Let $G=G_1\times\cdots\times G_k$ be a semisimple Lie group with all factors non-compact, and let $\Lambda<H$ be a cocompact lattice, with $G<H$. A left translation action $\rho$ o…

Let $G$) be a connected semisimple Lie group with finite center, let $\Gamma$ be a lattice in $G$, and let $Y$ be a compact homogeneous space of $G$. Let $K$ be a maximal compact s…

Let $G$, $\Gamma$, and $H$ be as in Margulis's orbit-closure conjecture, and let $\mu$ be an $H$-invariant, $H$-ergodic measure on $\Gamma\backslash G$. Suppose that for $\mu$-almo…

Let $G$, $\Gamma$, and $H$ be as in Margulis's orbit-closure conjecture, and let $\mu$ be an $H$-invariant, $H$-ergodic measure on $\Gamma\backslash G$. Suppose that for every…

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

Let $P(t)\in\mathbb{F}_q[t]$ be irreducible. For a lattice $\Lambda_0\in\mathcal{L}_2$, let $\mathbb{T}_P(\Lambda_0)$ be its $P(t)$-Hecke tree, and let…

Stable-perturbation conjecture. For every open cone $C\subseteq\mathfrak{a}$, every bounded $A_C$-orbit in $X_d$ contains an $A_C$-stable perturbation of a compact $A$-orbit. This…

Let $X_d=\mathrm{SL}_d(\mathbb{R})/\mathrm{SL}_d(\mathbb{Z})$ and let $A$ be the connected diagonal subgroup … An $A$-orbit is bounded if its image in $X_d$ is bounded. Margulis' c…

Let $d\geq 3$, let $X_d=\mathrm{SL}_d(\mathbb{R})/\mathrm{SL}_d(\mathbb{Z})$, and let $I_m,J_n$ be index sets. Choose nonzero vectors…

Let $d\geq 3$, let $X_d=\mathrm{SL}_d(\mathbb{R})/\mathrm{SL}_d(\mathbb{Z})$, and let $v^1\ne v^2$ be two nonzero vectors in the diagonal subalgebra of…

Let $(\alpha,\beta)$ be a pair of real numbers, and let $\epsilon>0$. Consider the system … A pair is called multiplicatively singular if, for every $\epsilon>0$, this system has a…

Let $(\alpha,\beta)$ be a pair of real numbers, and let $\epsilon>0$. The system … uses solutions $(p,r,q)\in\mathbb{Z}\times\mathbb{Z}\times\mathbb{N}$ for an unbounded set of…

Let $D$ be the diagonal group of $\operatorname{SL}_n(\mathbb{R})$ where $n\ge 3$. A $D$-orbit in $\operatorname{SL}_n(\mathbb{R})/\operatorname{SL}_n(\mathbb{Z})$ is relatively co…

HAW conjecture. For any $x\in X$, the set $E^+(x,\infty)$ is hyperplane absolute winning (HAW) on $\mathbf{P}^+(\mathfrak{g})$. This is the homogeneous-space counterpart of the pre…

Let $M$ be a locally symmetric space of non-compact type with finite Riemannian volume. For $p\in M$, let $S_p(M)$ denote the unit tangent sphere at $p$, and let $E^+(p,\infty)$ be…

Let $D$ be the set of diagonal matrices in $\mathrm{SL}_k(\mathbb R)$, where $k \geqslant 3$. Let $z \in \mathrm{SL}_k(\mathbb R)/\mathrm{SL}_k(\mathbb Z)$, and suppose that $Dz$ h…

Let $f$ be a product of $k$ linear forms in $\mathbb R^k$, where $k \geqslant 3$, and assume that $f$ is not proportional to a multiple of a polynomial with integer coefficients. L…

Let $G$ be a Lie group, let $X=G/\Gamma$ be a homogeneous space with $\Gamma<G$ cocompact and discrete, and let $f_0=L_a\circ\Psi$ be an affine diffeomorphism, where $a\in G$ and…

Fix an integer $d\geq 2$ and a prime power $q$. Let $\left[\mathcal{L}_d\right]$ be the space of homothety classes of lattices, let $\tilde{\mathcal{K}}^d$ be the ambient vector sp…

Let $d\geq 2$, let $G=\operatorname{SL}_d(\mathbb{R})$, let $\Gamma=\operatorname{SL}_d(\mathbb{Z})$, and let $X_d=G/\Gamma$, identified with the space of unimodular lattices in…