Line on torus as inspiration for finding counter example of homeomorphism theorem for topological group actions on topological space

Question

I have a textbook problem where I don't know how to start. I am invited to study a specific counter example (in a sense which remains to be clarified) illustrating the necessity of some conditions in the following theorem:

Let $G$ be a locally compact topological group which is also a countable union of compact sets and let $E$ be a locally compact topological space. Suppose $G$ acts continuously and transitively (only one orbit) on $E$. Then the quotient of $G$ with respect to the stabilizer of any point of $E$ is homeomorphic to $E$.

To find a counter example, I'm asked to examine the following problem :

Let $s$ be the canonical projection from $\mathbb{R}^n$ to $\mathbb{R}^n / \mathbb{Z}^n$, and let $D$ be a line through the origine in $\mathbb{R}^n$. Show that $s(D)$ is either compact or dense, according to whether $D \cap \mathbb{Z}^n \neq \{0\}$ or $D \cap \mathbb{Z}^n = \{0\}$.

I think I can do and understand the last point, but I don't see where I should seek a counter example. I have considered the action of $\mathbb{R}$ on the lines of the apparent counter example. In both cases I have the impression that the apparent counter example isn't a counter example to the theorem.

As a first step, I would be grateful for hints to understand what exactly might be the counter example here. Then, what is the necessary condition in the theorem which is not satisfied in the counter example ?

Answer 1

Hint: Put $G=\mathbb{R}$ and $E=\mathbb{T}^d=\mathbb{R}^d/\mathbb{Z}^d$ for $d=2$ ($\dagger$). Then $G$ is locally compact and $\sigma$-compact, and $E$ is locally compact. Let $\xi\in\mathbb{R}^d$ and suppose $G$ is acting on $E$ via translation in the $\xi$ direction: $t\curvearrowright x= x+t\xi$. Then this is a continuous action.

Thus the only issue could be regarding transitivity. Now note that any orbit of the translation action is the image of a translation of the line $\{r\xi\, |\, r\in\mathbb{R}\}\subseteq \mathbb{R}^d$ under the projection $\pi:\mathbb{R}^d\to\mathbb{T}^d$. Thus according to the auxiliary exercise you mention is

1. either an embedded circle in the torus xor
2. a dense subset in the torus,

so that the $G$ action on $E$ is never transitive (though in the second situation the action is said to be topologically transitive). (More generally one can use a Baire Category argument to show that no action of $\mathbb{R}$ is transitive on a topological manifold of dimension at least $2$.)

(See https://www.desmos.com/calculator/tk7uitvbpx for a humble interactive graph.)

In the first case the stabilizer subgroup of any point will be of the form $\rho_\xi \mathbb{Z}$ for some $\rho_\xi\in\mathbb{R}_{>0}$ uniquely determined by the vector $\xi$, and in the second case the stabilizer subgroup of any point will be $\{0\}$. Thus in the first case the quotient group is isomorphic to $\mathbb{T}$ and in the second case the quotient group is isomorphic to $\mathbb{R}$; clearly neither of these spaces are homeomorphic to $E$.

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($\dagger$) One can of course consider an arbitrary $d\in\mathbb{Z}_{\geq1}$. When $d=1$ the translation flow is either the trivial flow or a transitive flow. Also note that for $d>2$ the auxiliary exercise is not quite true, in that the dichotomy "compact xor dense" may fail (more precisely, the projections of lines may fail to be compact but also dense only in a subtorus of lower dimension) (e.g. consider the case $d=3$ and $\xi=(1,1,\sqrt{2})$); see the discussion at Dynamics on the torus for more details.