93 problems

Let $Q$ be an indefinite quadratic form in $n\ge3$ variables that is not proportional to a quadratic form with rational coefficients. Oppenheim conjecture. The set … is dense in…

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…

Raghunathan's conjecture. For any initial point $x\in X$, the orbit closure $\overline{U.x}$ is a periodic orbit $L.x$ of some subgroup $U\leq L\leq G$.

Let $X_n=\operatorname{SL}_n(\mathbb{R})/\operatorname{SL}_n(\mathbb{Z})$, and let $A$ denote the subgroup of positive diagonal matrices in $\operatorname{SL}_n(\mathbb{R})$. An…

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 $X$ be the quotient under consideration, let $h(t)$ denote the horocycle flow, let $p\in X$ be non-periodic, and let $\mu_X$ be the invariant measure. For a sequence $a_n$, the…

Let $\Gamma\subset PSL(2,\mathbb{R})$ be such that $X=\Gamma\backslash PSL(2,\mathbb{R})$ is not compact, and let $(h_t)$ be the horocycle flow acting on $X$. Margulis–Shah conject…

Kadyrov–Kleinbock–Lindenstrauss–Margulis conjecture. For every $h\in[m+n-1,m+n]$, there should exist a sequence of $a$-invariant probability measures $(\mu_k)_{k\geq 1}$ such that

Weiss's conjecture. The existence of divergent trajectories and of non-obvious divergent trajectories can be deduced from the relation between the dimension of $A$ and the…

Let the horocycle flow be the horocycle flow on a compact hyperbolic surface. Sinai's conjecture. The horocycle flow is mixing of all orders. This conjecture concerns higher-order…

Let $G'$ be a $Q$-group and let $Gamma'$ be an arithmetic lattice of $Q$-rank one. Let $T'$ act on $Gamma' \backslash G'$ by right multiplication by a diagonalizable element of…

Let $G=G_1\times\dots\times G_l$ be a direct product of simple groups, let $\Gamma$ be an irreducible lattice in $G$, and let $\rho_0$ be the regular representation of $G$ on…

Let $H$ be the diagonal split Cartan subgroup in $\operatorname{PGL}_n(\mathbb{R})$ for $n \geq 3$, acting on…

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 $G=\operatorname{SL}(3,\mathbb R)$, $\Gamma=\operatorname{SL}(3,\mathbb Z)$, and $X_3=\Gamma\backslash G$. Let $A<G$ be the diagonal subgroup, which is isomorphic to…

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…

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

Finite-measure and finite-component conjecture. The following statements hold: every locally finite $W$-invariant $W$-ergodic measure on $G/\Gamma$ is finite; if $\overline{Wx}=Fx$…

Margulis's generalized conjecture. Every finite $W$-invariant $W$-ergodic measure on $G/\Gamma$ should be homogeneous, and, when $G/\Gamma$ has finite volume, every $W$-orbit closu…

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 $d\geq 3$, 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…