Timeline for growth rate of $\mathbb{Z}^2\rtimes_{\sigma} \mathbb{Z}$?
Current License: CC BY-SA 3.0
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
Commonmark migration
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Aug 13, 2013 at 19:14 | vote | accept | Jiang | ||
| Aug 13, 2013 at 2:47 | comment | added | Steve D | @Jiang: For your group to be (virtually) nilpotent, $\sigma(a)$ must fix a point (otherwise the center of $G$ would be trivial). You can also check this is a sufficient condition (quotient by the fixed subgroup, and check it is (virtually) abelian). | |
| Aug 11, 2013 at 22:17 | comment | added | Jiang | @Lee, especially, is the lemma 1 in Milnor's paper useful in our situation? | |
| Aug 11, 2013 at 22:14 | comment | added | Jiang | @Lee, in your answer, you mentioned Milnor's paper, I checked it, but it is still not clear to me how to relate the nilpotentness of $G$ to the property of $\sigma(a)$, could you give more hints? | |
| Aug 11, 2013 at 16:02 | comment | added | Lee Mosher | The theorem that $H_3$ and other (virtually) nilpotent groups have polynomial growth is a theorem of Milnor, with the exact degree of polynomial growth computed by Bass. Gromov's theorem is the converse: every group of polynomial growth is virtually nilpotent. | |
| Aug 11, 2013 at 15:57 | answer | added | Lee Mosher | timeline score: 4 | |
| Aug 11, 2013 at 14:35 | history | asked | Jiang | CC BY-SA 3.0 |