Timeline for Can one pose a Toeplitz index problem associated to a discrete group?
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| Jul 18, 2010 at 15:09 | comment | added | Paul Siegel | @Victor: I did consider trying to update the page as you say, but I am nervous because all I really know anything about is the index theory for Toeplitz operators (and even there I am by no means an expert). There are huge swaths of the theory that I have at most heard of. So I don't think I could actually enlarge the page beyond what would deserve to be called a stub. | |
| Jul 18, 2010 at 13:04 | comment | added | Victor Protsak | Re "let me provide a little background since the Wikipedia page on this stuff is sorely lacking": one of the great advantages of Wikipedia is at it's an encyclopaedia that anyone can edit! Thus if you realize that some important information is missing and you have expertise to fill it in, by all means, do it! Note that your contribution at WP will have a lot more impact than a few lines prefacing a question in MO. | |
| Jul 18, 2010 at 2:55 | comment | added | Paul Siegel | Hmmm... good point. I'm tempted to abandon my initial expectation that this might be easier for hyperbolic groups. In light of your comment, maybe it's worth thinking about amenable groups instead. | |
| Jul 17, 2010 at 2:39 | comment | added | user2412 | Paul, just as a comment, for a hyperbolic group there is no invariant measure on the usual Gromov boundary. This follows from the fact that such a group is non-amenable, while the action on the Gromov boundary is topologically amenable. | |
| Jul 16, 2010 at 20:15 | history | edited | Paul Siegel |
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| Jul 16, 2010 at 20:12 | comment | added | Paul Siegel | I should also mention that there is a Toeplitz index theorem for complete Riemannian manifolds with boundary which has a slightly different flavor from what I outlined. In that case one builds favorable boundary behavior into the geometry instead of the analysis by stipulating a certain compatibility between the large-scale properties of the metric on the interior and the metric on the boundary. Instead of using Hardy spaces, one builds the Toeplitz operators by projecting onto the kernel of the Dirac operator. Perhaps this idea is the right one to try to generalize. | |
| Jul 16, 2010 at 20:04 | comment | added | Paul Siegel | I suggested hyperbolic because it seems as though there are often fewer subtleties associated with compactifying negatively curved spaces (many of the natural candidates are the same). But maybe what is needed to get things off the ground is a simplifying assumption which suggests a good candidate for the space that I called $F(\Gamma)$. | |
| Jul 16, 2010 at 19:20 | comment | added | Yemon Choi | Also, is there a geometric reason that you went straight to the case of hyperbolic groups? I guess the disc is something with natural hyperbolic metric... but then I can't think, off the top of my head, what a good discretized version of holomorphic might be. Perhaps some kind of submean/Harnack inequality, or growth conditions? | |
| Jul 16, 2010 at 19:15 | comment | added | Yemon Choi | While this doesn't answer your question: generalizations of the Toeplitz-operator framework have been looked at by various people (Murphy, Douglas) but they seemed to be looking at ordered abelian groups. | |
| Jul 16, 2010 at 19:11 | comment | added | Yemon Choi | @Paul: no, but if you think one would be helpful then go for it - there are a few questions I've seen which could have been tagged thus, and peple I've seen on the site who might use such tags | |
| Jul 16, 2010 at 18:15 | comment | added | Paul Siegel | Is there a "geometric group theory" tag? | |
| Jul 16, 2010 at 18:15 | history | asked | Paul Siegel | CC BY-SA 2.5 |