Timeline for Mostow Rigidity Theorem and reconstruction from fundamental group
Current License: CC BY-SA 4.0
10 events
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| Apr 26, 2020 at 16:13 | comment | added | Toffee | @JeanRaimbault As for other isolated points, I don't know an example off the top of my head. You could probably build one with a dominant map between closed hyperbolic $3$-manifolds that isn't a covering. Such an example is probably sufficiently complicated that cranking out the character variety isn't practical though. There are known examples of dominant maps between knot complements, so you see the canonical component of the target knot complement in the character variety of the other one. I can't remember one off-hand, but papers by Boyer and Zhang are a likely source. | |
| Apr 26, 2020 at 16:13 | comment | added | Toffee | @JeanRaimbault That's right about local rigidity, except for the fact that local rigidity fails for nonuniform lattices in $\mathrm{SL}_2(\mathbb{C})$. Local rigidity only holds once you deem the parabolic subgroups must stay parabolic (i.e., traces of peripheral elements are $\pm 2$). | |
| Apr 26, 2020 at 15:03 | comment | added | Jean Raimbault | Have the isolated points that do not come from the monodromy representation or its Galois conjugates ever been looked at? I can't even think of an example. | |
| Apr 26, 2020 at 14:56 | comment | added | Jean Raimbault | the rigidity result which gives discrete+faithful => isolated in character variety would be local (Calabi--Weil) rigidity rather than strong (Mostow) rigidity. | |
| Apr 25, 2020 at 15:17 | vote | accept | Cameron Zwarich | ||
| Apr 25, 2020 at 15:17 | comment | added | Cameron Zwarich | Thanks for this great answer. It's more than what I was expecting to get. | |
| Apr 24, 2020 at 17:25 | history | edited | Toffee | CC BY-SA 4.0 |
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| Apr 24, 2020 at 17:14 | history | edited | Toffee | CC BY-SA 4.0 |
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| Apr 24, 2020 at 17:01 | history | edited | Toffee | CC BY-SA 4.0 |
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| Apr 24, 2020 at 16:54 | history | answered | Toffee | CC BY-SA 4.0 |