Timeline for Representation theory and topology of Teichmüller space
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
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| Dec 19, 2023 at 1:29 | vote | accept | Kenny S | ||
| Dec 18, 2023 at 6:28 | comment | added | Kenny S | @AndyPutman, I try to write down the details of these, then I encounter some questions. "so the set of homotopy classes of homeomorphisms from 𝑆 to 𝑆′ is the same as the set of conjugacy classes of isomorphisms between their fundamental groups. " I want to derive the homotopy classes of isometry and not just homeomorphism, when I want to use the Prop 1.B.9 in Hatcher's book, which says a proposition of K(G,1). | |
| Dec 15, 2023 at 21:51 | answer | added | Sam Nead | timeline score: 3 | |
| Dec 15, 2023 at 3:36 | comment | added | Kenny S | @AndyPutman, Very good explanation! By the way, how can I compute the rep space? Why it's $char_{2}\pi_{1}(S)$? | |
| Dec 15, 2023 at 3:08 | comment | added | Andy Putman | @KennyS: If $S$ and $S'$ are two surfaces of genus $g \geq 1$, then they're both $K(\pi,1)$'s for their fundamental group, so the set of homotopy classes of homeomorphisms from $S$ to $S'$ is the same as the set of conjugacy classes of isomorphisms between their fundamental groups. Thus a marking of a surface of genus at least $1$ is the same as an identification of its fundamental group with that of your fixed reference surface (up to conjugacy). | |
| Dec 15, 2023 at 2:53 | comment | added | Kenny S | @AndyPutman, thank you for your comment, the first half of what you said I can fully understand, but why "$\pi_{1}(S)$ is identified with $\Gamma$" can be related to the markings? This fact seems established anytime when we have a universal covering and identify the fundamental group with deck transformation group. I can't see where the marking appears. May you explain in details when you have time? As an answer I'd prefer | |
| Dec 15, 2023 at 2:14 | comment | added | Andy Putman | If you have a marked Riemann surface $S$ of genus $g \geq 2$, the uniformization theorem says that its universal cover is biholomorphic to the unit disc $D$. It's a fun exercise in complex analysis that the group of biholomorphisms of $D$ is $\text{PSL}_2(\mathbb{R})$. This shows that $S \cong D / \Lambda$ where $\Lambda$ is a discrete subgroup of $\text{PSL}_2(\mathbb{R})$. The marking on $S$ lets you identify $\pi_1(S)$ with $\Lambda$ (up to conj due to basepts), so we get a point in your rep space. Reversing this process gives you the inverse map from the rep space to Teichmuller space. | |
| Dec 15, 2023 at 2:05 | history | edited | Kenny S | CC BY-SA 4.0 |
added 685 characters in body
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| Dec 15, 2023 at 1:45 | comment | added | Kenny S | @LSpice, thank you so much for your edition of my question, I will remember next time to inline the image and make the question more self-contained! The note is written by Jing Tao and the name of the note is "Introduction to Teichmuller spaces", I somehow downloaded it but right now I can't find the original links, I will add the link of note later when I find it! | |
| Dec 15, 2023 at 1:43 | comment | added | Kenny S | @LeeMosher, sorry, I will add the definition in the question later. There are many equivalent definitions, I know what you said "In some treatments...", the definition you gave is indeed what higher Teichmüller theory comes from. But as a beginner, I wonder how the definition you gave(discrete faithful representations) agrees with the definitions that uses "marking Riemannian surface" (as I added later in the question) | |
| Dec 14, 2023 at 14:56 | comment | added | LSpice | What is the note? I also took the time to inline the image, and correct a typo ($6g - g$ was supposed to be $6g - 6$). Whenever possible, it is best to make your posts self-contained, not relying on an image (which, among other things, is not searchable) for textual content. | |
| Dec 14, 2023 at 14:56 | history | edited | LSpice | CC BY-SA 4.0 |
Inlining image
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| Dec 14, 2023 at 14:16 | comment | added | Lee Mosher | Your question is hard to answer without knowing how you want to define Teichmüller space, because your question is very close to the level of the very definition of $T_g$. In some treatments, $T_g$ is actually defined to be the subset of $Hom(\pi_{1}({S_g}),PSL_{2}(\mathbb{R}))/PSL_{2}(\mathbb{R})$ represented by faithful representations $\pi_{1}({S_g}) \mapsto PSL_{2}(\mathbb{R})$ with discrete image. | |
| Dec 14, 2023 at 10:17 | history | asked | Kenny S | CC BY-SA 4.0 |