6 problems

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 $g\geq 1$, let $\alpha=(\alpha_1,\dots,\alpha_n)$ be a partition of $2g-2$, let $\mathcal H_1(\alpha)$ be the space of unit-area surfaces in the corresponding stratum of abelia…

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 $G$ be a Lie group, let $\Gamma$ be a discrete subgroup of $G$, let $\{u_t\}_{t\in\mathbb{R}}\subseteq G$ be a unipotent one-parameter subgroup, and let $x=g\Gamma\in G/\Gamma$…