Timeline for Representation theory and topology of Teichmüller space
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 19, 2023 at 7:58 | comment | added | Kenny S | I have post another detailed question under your advice, see here: mathoverflow.net/q/460659/516181, I welcome your answers. | |
| Dec 19, 2023 at 1:35 | comment | added | Kenny S | Okay I have cheaked your answer. I know the FN coordinates, but I think it's another way to derive the dimension, as you saw at the very beginning of my question, I assumed that the representation way to express the Teichmuller space can compute the dimension purely algebraically, without use of FN coordinates. Now I feel the representation definition is more like a "formal" or "abstract" definition with no practical use. | |
| Dec 19, 2023 at 1:29 | vote | accept | Kenny S | ||
| Dec 18, 2023 at 10:46 | comment | added | Sam Nead | I should have said “one short answer to the question”, as there are other computations of the dimension. | |
| Dec 18, 2023 at 9:42 | comment | added | Sam Nead | But the short answer to the question "why is the dimension 6g - 6?" is as follows. The surface has 3g - 3 curves in any pants decomposition. Every curve in a pants decomposition contributes two real parameters: its length and its twist. See the so-called Fenchel–Nielsen coordinates for Teichmuller space (which is homeomorphic to a component of the character variety). | |
| Dec 18, 2023 at 9:37 | comment | added | Sam Nead | Just a few points of etiquette: if my answer answers your original question, then you should accept it (by clicking on the "check" mark). If you have another question (for example, how to compute the dimension of the character variety), then you should ask that in a new question. | |
| Dec 17, 2023 at 0:54 | comment | added | Kenny S | Thank you for your answer! Forgive me that I'm not familiar with "character variety", can you tell me how to calculate the dimension of it for arbitrary fundamental groups of Riemann Surfaces of genus g? why it's 6g-6? I search this on wiki and the calculation is given only in special cases. Maybe a reference includes this calculation is enough if you think the computation is too complex to type here. | |
| Dec 15, 2023 at 21:51 | history | answered | Sam Nead | CC BY-SA 4.0 |