Timeline for When are root hyperplanes locally finite?
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| Aug 16, 2011 at 9:38 | comment | added | A. Pascal | I thought more about the above example with a $\hat{A_{1}}$ inside an $\mathbb{R}^{2,1}$. This is revealing. Thank you. For other readers, and for concreteness, a good model in which to understand the above is the following: $L$ is a lattice with integral basis $u_{1},u_{2},u_{3}$ and pairings $\langle u_{i},u_{i} \rangle=2$, $\langle u_{i},u_{j} \rangle=-2$ if $|i -j|= 1$, and $\langle u_{i},u_{j} \rangle=0$ otherwise. Then the 'singular subspace' is spanned by $u_{1},u_{2}$. Inside there is an $\hat{A_{1}}$. Singular means that the quadratic form restricted to this subspace is degenerate. | |
| Aug 10, 2011 at 8:46 | comment | added | A. Pascal | Sorry. I wasn't clear. By 'discrete group' I meant 'group acting discretely', which is not the case for infinite groups acting on compact Hausdorff spaces (like $PGL_{2} (\mathbb{Z})$ on $\mathbb{P}^{1} (\mathbb{R})$, as you point out). In the linear case under question, though, I was considering infinite groups acting on linear (in particular non-compact) spaces, and I'm pretty sure they are acting discretely, being subgroups of $GL_{n}(\mathbb{Z})$. So I think there should be some fundamental domain, even if it is impossible to understand. But maybe I am missing something. | |
| Aug 9, 2011 at 5:42 | comment | added | S. Carnahan♦ | Fundamental domains don't necessarily exist when you have non-compact groups of symmetries, because orbits of points can exhibit accumulation. The standard example is the action of $PGL_2(\mathbb{Z})$ on $\mathbb{P}^1 (\mathbb{R})$ by Möbius transformations. In the case at hand, we are considering discrete subgroups of a noncompact orthogonal group acting on a space of substantially lower dimension. | |
| Aug 8, 2011 at 17:28 | comment | added | A. Pascal | About the last comment on signature $(m,n)$. Shouldn't any discrete group admit some fundamental domain? | |
| Aug 8, 2011 at 17:27 | vote | accept | A. Pascal | ||
| Aug 8, 2011 at 19:04 | |||||
| Aug 8, 2011 at 15:34 | history | answered | S. Carnahan♦ | CC BY-SA 3.0 |