Double Coset Closed

Question

Let $G$ be a locally compact group and $H$ a closed subgroup. Under what conditions can we say that the double cosets $H\cdot x \cdot H$ are closed? Is this always true? I am interested mainly in the case when $H$ is discrete.

Answer 1

Closeness of cosets is definitely false in general even for discrete subgroups of Lie groups. For instance, suppose that $M=G/H$ is compact. The group $H$ acts on $M$ via left multiplication. Closeness of your double closets just says that the orbits of the action on $M$ are closed. However, if, say, $G=SL(2, {\mathbb R})$ and $H$ is discrete, then most orbits will be dense in $M$.

Edit: Consider a discrete subgroup $H$ of $G$ so that $M=G/H$ is compact. Without loss of generality (by conjugating $H$ via an element of $G$) we may assume that $H$ contains an element of infinite order $g$ which is a diagonal matrix with positive diagonal values. I claim that the action of $g$ on $M$ via let multiplication is ergodic, which would imply that almost every $<g>$-orbit in $M$ is dense in $M$ (the same, of course, will be also true for the entire group $H$ since it contains $<g>$). Indeed, the action of $g$ on $M$ is an infinitely differentiable Anosov diffeomorphism, since the action of the diagonal group $$ D= \left\{ \left[\begin{array}{cc} e^t&0\\ 0&e^{-t} \end{array}, \right], t\in {\mathbb R}\right\} $$ (containing $g$) is Anosov (see e.g. http://en.wikipedia.org/wiki/Anosov_diffeomorphism for the definition and explanations): The action of the diagonal group is also called the "geodesic flow" on $M$, it preserves a natural volume form on $M$. Anosov proved in his thesis (in the 1960s) that every $C^2$-smooth volume preserving Anosov diffeomorphism of a smooth compact manifold is ergodic. (I am guessing, but this is only a guess, that Anosov did not call Anosov diffeomorphisms "Anosov".) I think, you can find a proof of this ergodicity result in the book by M.Brin (father of Sergei Brin of google fame) and G.Stuck "Introduction to dynamical systems". Gergodocity of the action of $g$ also follows from ergodicity of the geodesic flow on $M$ (with a bit more thought); proof of this ergodicity theorem due to H.Hopf should be also in the Brin-Stuck book.