Timeline for Existence of properly discontinuous and cocompact action
Current License: CC BY-SA 4.0
9 events
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| Aug 20, 2022 at 22:36 | comment | added | HJRW | @SMS: Regarding your second comment, the Svarc--Milnor lemma implies that $N$ is quasi-isometric to $\Gamma$ and so, by a well known result of Hopf, must have 0, 1, 2 or $\infty$ ends. (And each of these cases is fairly easy to realise.) | |
| Aug 20, 2022 at 22:33 | comment | added | HJRW | @SMS: Regarding sufficient conditions, the first non-trivial (hyperbolic) case is in dimension 3, and by Theorem B we may assume that $\pi_1M$ is a surface group. The question is then whether or not $\pi_1M$ is a geometrically finite or geometrically infinite Kleinian group. I believe (perhaps someone else can confirm) that all geometrically infinite hyperbolic surface groups can be embedded as normal subgroups of lattices, whence the relevant group action will exist. I think nothing is known above dimension 3. | |
| Aug 20, 2022 at 19:25 | comment | added | SMS | In particular, I was thinking of the following little problem: can one find a closed Riemannian surface $M$ and a complete non-compact Riemannian covering space $N$ such that $M = N /\Gamma$, and $N$ has prescribed number of ends? Intuitively this seems doable, but at the moment, I cannot come up with a construction. | |
| Aug 20, 2022 at 18:43 | comment | added | SMS | I am really interested. Do you know of any more sufficient conditions? Your Theorem B and Theorem C are necessary conditions. | |
| Jul 20, 2022 at 18:32 | history | edited | HJRW | CC BY-SA 4.0 |
added 5 characters in body
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| Jul 20, 2022 at 18:26 | history | edited | HJRW | CC BY-SA 4.0 |
Corrected theorem C
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| Jul 20, 2022 at 15:11 | history | edited | LSpice | CC BY-SA 4.0 |
Title of answered question
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| Jul 20, 2022 at 13:59 | history | edited | HJRW | CC BY-SA 4.0 |
Added reference to Paulin's theorem
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| Jul 20, 2022 at 12:32 | history | answered | HJRW | CC BY-SA 4.0 |