6 problems

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

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…

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 $D$ 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…