Timeline for Involution-free Trees are Asymmetric: Reference request
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Apr 27, 2011 at 15:31 | comment | added | darij grinberg | Oh, and I should have read the OP. Headdesk. | |
| Apr 27, 2011 at 15:27 | history | edited | darij grinberg | CC BY-SA 3.0 |
added 66 characters in body
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| Apr 27, 2011 at 15:27 | comment | added | darij grinberg | You are right about Lemma 2; editing the post. | |
| Apr 27, 2011 at 15:24 | comment | added | darij grinberg | Thanks! And Serre gives a very nice strengthening: If the diameter of the tree is even, it has a vertex which is fixed under every automorphism. If the diameter of the tree is odd, it has an edge which is fixed under every automorphism. | |
| Apr 27, 2011 at 15:22 | comment | added | Tom De Medts | @darij: Your proof of Lemma 2 looks overly complicated. Here is an easier way to phrase your argument. Let $g$ be an arbitrary automorphism of the tree, and assume that $g$ fixes every vertex at level $n$ but moves some vertex $v$ at level $n+1$. Then the subtrees emanating from $v$ and $vg$ are isomorphic. Hence there is an involution of the tree interchanging the subtrees emanating from $v$ and $vg$, fixing all vertices of level $\leq n$ and also fixing all vertices for which the minimal path to the root does not contain $v$ or $vg$. | |
| Apr 27, 2011 at 15:20 | history | edited | darij grinberg | CC BY-SA 3.0 |
typos fixed
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| Apr 27, 2011 at 15:12 | comment | added | Tom De Medts | A possible reference for Lemma 1 is Jean-Pierre Serre's wonderful book "Trees", p.20, Corollary to Proposition 10. | |
| Apr 27, 2011 at 15:06 | history | answered | darij grinberg | CC BY-SA 3.0 |