Raghunathan's conjecture

Remarks

The statement was conjectured by M. S. Raghunathan and proved by Marina Ratner in 1991. Her proof derives it from her classification of the invariant measures of unipotent flows, one of Ratner's theorems. Before that, Margulis studied $H = \mathrm{SO}(2, 1)^\circ$ acting on $\mathrm{SL}_3(\R)/\mathrm{SL}_3(\Z)$. He proved that every relatively compact $H$-orbit there is closed, and deduced the Oppenheim conjecture.

Example: for $G = \R^2$, $\Gamma = \Z^2$ and $H$ a line through $0$ of slope $\alpha$, all elements are unipotent. If $\alpha$ is rational, each orbit of $H$ on the torus is a closed circle, and $L = H$. If $\alpha$ is irrational, each orbit is dense, and $L = G$.

The unipotent hypothesis is necessary: orbits of the diagonal group in $\mathrm{SL}_2(\R)/\mathrm{SL}_2(\Z)$ can have closures that are not orbits of a subgroup.

Going deeper

For a student

The theorem is the topological shadow of Ratner's measure classification. It is the main tool for questions about the values of polynomials at integer points. The first such question was the Oppenheim conjecture. Let $Q$ be a nondegenerate indefinite real quadratic form in $n \ge 3$ variables, not a multiple of a form with rational coefficients. Then the values $Q(\Z^n)$ are dense in $\R$. The theorem shows that orbit closures of unipotent flows behave like closures of linear flows on tori, but on noncommutative spaces.