Timeline for Fundamental group of a compact space form.
Current License: CC BY-SA 2.5
10 events
| when toggle format | what | by | license | comment | |
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| Mar 26, 2010 at 12:23 | history | edited | Matthew Stover | CC BY-SA 2.5 |
added 2 characters in body
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| Mar 26, 2010 at 12:14 | history | edited | Matthew Stover | CC BY-SA 2.5 |
Added a condition
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| Mar 26, 2010 at 10:02 | comment | added | Tom Church | For fundamental groups of negatively curved manifolds, the answer above should say "compact hyperbolic manifolds never contain a $\mathbb{Z}\oplus\mathbb{Z}$". You can see this directly without appealing to the fact that they are Gromov-hyperbolic from the fact that centralizers are virtually cyclic. To see this, note that a hyperbolic isometry has two fixed points on the boundary at infinity. The centralizer of a given hyperbolic isometry must preserve these fixed points, and so sits inside $\mathbb{R}\times SO(n-1)$. A discrete subgroup of this group is virtually $\mathbb{Z}$. | |
| Mar 26, 2010 at 9:57 | comment | added | Tom Church | For hyperbolic groups, a $\mathbb{Z}\oplus\mathbb{Z}$ subgroup has triangles which are as fat as triangles in the Euclidean plane, which violates the thin triangles condition which is the definition of hyperbolic group. | |
| Mar 26, 2010 at 5:57 | comment | added | Pete L. Clark | How do you know that hyerbolic groups are not virtually abelian? I mean, I certainly believe it to be true -- I have studied Fuchsian groups, where this follows from the explicit presentation of the group in terms of its signature -- but what technique or fact are you using to see this? (You could use Milnor's theorem, as above, but maybe you have something else in mind?) | |
| Mar 25, 2010 at 20:31 | comment | added | engelbrekt | Thanks for the quick answer! I am only allowed to check one answer as accepted, but morally I accept yours too, of course. | |
| Mar 25, 2010 at 20:27 | vote | accept | engelbrekt | ||
| Mar 25, 2010 at 20:27 | |||||
| Mar 25, 2010 at 20:26 | vote | accept | engelbrekt | ||
| Mar 25, 2010 at 20:26 | |||||
| Mar 25, 2010 at 20:25 | history | edited | Matthew Stover | CC BY-SA 2.5 |
Gave better justification
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| Mar 25, 2010 at 20:19 | history | answered | Matthew Stover | CC BY-SA 2.5 |