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The following are my impressions from playing with a line of irrational slope $\gamma$ in the standard torus $S^1\times S^1=\mathbb{R}^2/\mathbb{Z}^2$, $\mathbb{R}\hookrightarrow \mathbb{R}\mathbb\times\gamma\mathbb{R}$. The topology induced on $\mathbb{R}$ by restricting open sets of $T^2$ is strictly coarser than the usual order topology on $\mathbb{R}$. In fact, the basis consists of countable unions of neighborhoods. The topology has a sort of "fractal" feel to it: for any integer $N$, $\epsilon>0$, there exists some $x>N$ with $d(x,0)<\epsilon$. In fact, for a fixed $\epsilon$, the ball of radius $\epsilon$ about $0$ "unwraps" to intervals which, loosely speaking, approach length $2\epsilon$ as one approaches $\infty$. The spacing of the intervals also seems like it is regular and governed by $\gamma$ and $\epsilon$. That seems pretty neat to me.

Based on this and motivated by idle curiosity, I'm asking for (preferably modern) references which contain a systematic study of non-closed subgroups of Lie groups, as opposed to the usual off-hand mention that they exist and are not interesting. I've turned up a couple of papers in the early '90s, but none of them are quite what I'm looking for. The closest is probably Virgos, "Non-closed Lie Subgroups of Lie Groups" (Annals of Global Analysis and Geometry 11, 1993) Content: Homotopy obstructions to existence of connected nonclosed Lie subgroups, construction of some examples. But this is still not quite the sort of analysis I'm looking for.

Apparently there was a bit of work done in the '40s and '50s on the topic; names like Malcev and Van Est turn up in the "modern" references. I've also run across a paper by Goto.

To more precisely restate my question:

Is there any modern work studying non-closed Lie subgroups of Lie groups, or any survey papers or syntheses of older results? By "modern" I mean from the last twenty years (aside from the already-mentioned papers by Virgos and Kubarski)

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Irrational flows on the torus are well-studied (and famous) examples in noncommutative geometry. So it might be worth looking at the noncommutative geometry literature to see if there is anything like what you're asking. –  Jonathan Feb 9 '12 at 4:31
    
@Jonathan -- Thanks! I'll check into that, although that's not exactly what I'm looking for. I'm more after any systematic study of non-closed Lie subgroups in general. (I'll take what I get, though!) [Edit - Bumping the question to get a few more eyes looking over it.] –  Neal Feb 20 '12 at 4:00

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