---
tag: ME4ZSRAT5Z
kind: statement
role: theorem
name: Raghunathan's conjecture
msc: [37A17, "22E40"]
uses: [7JBRGCVJRB, BTYF5Y5PFY, 0QEAJB91GZ, 0YQPZB5Q1G]
about: [BTYF5Y5PFY]
sources:
  - url: https://en.wikipedia.org/wiki/Ratner's_theorems
---

## Statement

Let $G$ be a connected [[Lie group|7JBRGCVJRB]], and let $\Gamma$ be a [[lattice|BTYF5Y5PFY]] in $G$.
Let $H$ be a connected closed subgroup of $G$ generated by one-parameter subgroups $(u_t)$ such that each $\mathrm{Ad}(u_t)$ is unipotent.
Then for every $x \in G/\Gamma$ there is a closed subgroup $L$ of $G$ containing $H$ with the following property.
The [[closure|0QEAJB91GZ]] of the [[orbit|0YQPZB5Q1G]] $Hx$ equals the orbit $Lx$, and $Lx$ carries an $L$-invariant probability measure.

## 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|3MXR03096V]].
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.

## Explanations

### L1 human

Lines of rational slope on a torus close up, and lines of irrational slope fill the torus.
The conjecture says that the same simple dichotomy, in a more general form, holds for orbits of unipotent groups on all spaces of the form $G/\Gamma$.
The closure of an orbit is never a messy set: it is always a smooth homogeneous piece of the space.

### L2 human

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.