Timeline for sub-tori of a torus, generated by 1-dimensional subgroup
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 12, 2010 at 19:28 | vote | accept | CuriousUser | ||
| May 12, 2010 at 19:10 | comment | added | CuriousUser | Thanks, i got what you meant it after I read Ekedahl's post! | |
| May 12, 2010 at 18:39 | comment | added | BCnrd | I was thinking more algebraically, using closure of subgroup generated by $\overline{v}$ rather than the 1-parameter subgroup passing through $\overline{v}$. That is, I made the same misreading as Ekedahl. The example I gave in my comment is a point generating a subgroup whose closure is the first circle factor and does not contain $\overline{v}$. Ekedahl's answer also shows that I missed the possibility that the $\mathbf{Q}$-span of the $v_i$'s contains 1, in which case the dimension drops by 1 from the guess; in fact, my example using $(\sqrt{2}, 1/2)$ demonstrates that! Mea culpa. | |
| May 12, 2010 at 17:48 | comment | added | CuriousUser | Why is the closure not connected? I mean, by definition my one-parameter group (call it G) is dense in the closure, and it is connected. If the closure consisted of 2 or more connected components A and B, and G is either in A or B. Say G is in A. but since A and B are both open and closed, the closure of G must be contained in A and cannot be A\cup B... Again, am i missing something? | |
| May 12, 2010 at 17:27 | answer | added | Torsten Ekedahl | timeline score: 7 | |
| May 12, 2010 at 17:11 | comment | added | BCnrd | Nice question (& correct guess). Beware that the closure may not be connected (consider the point $(\sqrt{2},1/2)$ in $T^2$), so the torus you want is the identity component of the closure. Replacing $\overline{v}$ with a nonzero integral multiple is then harmless. Make integral change of coords and drop $n$ if necessary to get to case when $v_i$'s are lin. indep. over $\mathbf{Q}$. Then $\overline{v}$ generates dense subgp of $T^n$: if not get $T^1$-quotient killing it. Change coords (and pass to multiple) so quotient map is projection to 1st factor, contradicting lin. indep. hypothesis. | |
| May 12, 2010 at 16:28 | history | asked | CuriousUser | CC BY-SA 2.5 |