The inclusion $H\Gamma/\Gamma \hookrightarrow G/\Gamma$ is proper for closed subgroup $H$ and discrete subgroup $\Gamma$?

Question

Let $G$ be a Lie group (maybe enough to assume a locally compact Hausdorff group) and $H$ be a closed subgroup and $\Gamma$ be a discrete subgroup. I wonder if the following statement is true:

The inclusion $H\Gamma/\Gamma \hookrightarrow G/\Gamma$ is proper

My attempt: Let $K$ be a compact subset of $G/\Gamma$. Since $K$ is closed, we have $K=F\Gamma/\Gamma$ for some closed subset $F$ of $G$ (by definition of closed sets in quotient topology). But I don't know if $F$ can be made into a compact subset.

Answer 1

Perhaps, the easiest counter-example to this conjecture is $G=\mathbb R$, $H=\mathbb Z< G$, and $\Gamma< G$ generated by $\sqrt{2}$ (or any irrational real number for this matter). Then $H\Gamma$ is dense in $G$ and $(H\Gamma)/\Gamma$ is a nondiscrete (non-locally compact) topological group. The map (actually, a continuous homomorphism) $(H\Gamma)/\Gamma\to G/\Gamma$ is not proper.

The missing assumption (that Raghunathan has, of course) is that $H\cap\Gamma$ is a lattice in $H$. If assume that $H\cap\Gamma$ is a lattice in $H$ then you get the desired conclusion.

Answer 2

This is actually a hard theorem. See Theorem 1.13 in Discrete Subgroups of Lie Groups by M. S. Raghunathan. $H\Gamma/\Gamma \cong H/H\cap \Gamma$ (homeomorphic)