Automorphisms on disc | Mathematics | Page 1

delicate vale May 21, 2024, 10:50 AM

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Our professors asked us: "Is there an automorphism of the disc such that it has no fixed points?"
After this he showed that there is an isomorphism between (Aut(Disc), ○) ~ (Aut(SuperiorSemiPlane), ○) and the semiplane really does have automorphisms that have no fixed points, i.e. horizontal translation.
So this is supposed to porve that there exists back an automorphisms of the disc that similarry doesnt have a fixed point.

Question: Why would the fact that there are autom with no fp in the semiplane mesn that there are also auto with no fp on the disc? Does the isomorphism between the group preserve the "*no fixed point-ness"?

lethal zincBOT May 21, 2024, 10:50 AM

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delicate vale Our professors asked us: "Is there an automorphism of the disc such that it has ...

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delicate vale May 21, 2024, 10:51 AM

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If so, why?

tender flame May 21, 2024, 2:57 PM

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What is the semi plane Ive never heard that notation before

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The standard results regarding unit disc automorphisms are proved using the Schawrz lemma see here https://www.ma.imperial.ac.uk/~dcheragh/Teaching/2016-F-GCA/2016-F-GCA-Ch2.pdf

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And if there is some isomorphism between maps as your professor seems to suggest (which i dont know anything about) then yes that gives you a correspondence between the algebras of automorphisms in one space and the other so your actual question is just answers as yes simply by definition of isomorphisms.

delicate vale May 22, 2024, 9:12 AM

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How come its by definition of isomprphism?

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Why would isomorphisms preserve the no fixed pointness propriety?

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@tender flame

tender flame May 22, 2024, 1:15 PM

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Because fixed points would be preserved via the isomomorphism?

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I mean look I don't know what result your prof actually did if you can send the proof I can read through it

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Cause I haven't seen this exact result before

delicate vale May 22, 2024, 2:43 PM

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The isomorphism between autD and autH?

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Or the existence of automorphisms with no fixed points on D and H?

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There isn't anything else to the proof except what I said above

#Automorphisms on disc