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Results tagged with riemann-surfaces
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user 317
Riemann surfaces(Riemannian surfaces) is one dimensional complex manifold. For questions about classical examples in complex analysis, complex geometry, surface topology.
9
votes
Teichmuller theory and moduli of Riemann surfaces
I'll discuss things which are more applications of the mapping class group to moduli space rather than Teichmuller theory per se, but of course this is all tightly connected.
One of the big applicati …
- 44.8k
4
votes
Accepted
Can we prove $ Aut(S_g) , g \geq 2 $ is finite in the following way ?
The answer is yes. It's a little easier if we consider oriented geodesics (which is fine for your argument). Let $\alpha$ and $\beta$ be two oriented simple closed curves on $S_g$ that intersect onc …
- 44.8k
6
votes
Accepted
Are transversely immersed PL surfaces Riemann surfaces?
This question seems very confused. It is true that every PL surface can be given a canonical smooth structure. It is also true that a surface $X$ that is smoothly immersed in $\mathbb{R}^n$ can be g …
- 44.8k
18
votes
Accepted
Are mapping class groups of orientable surfaces good in the sense of Serre?
This is an open and probably very difficult question. There have been purported proofs (for instance, this one), but they have all had fatal flaws.
The mapping class group is definitely not virtuall …
- 44.8k
7
votes
Translation surfaces
There is a huge literature, and I'm not sure exactly what you are looking for. That being said, Masur-Tabachnikov's survey "Rational billiards and flat structures" and Masur's survey "Ergodic Theory …
- 44.8k
2
votes
Basic Questions about Teichmuller's theorem/quadratic differentials
I don't have time to answer your substantive questions, but I can recommend two very good sources for your 4th question :
Quasiconformal maps & Teichmüller theory by Fletcher-Markovich
Teichmüller T …
- 44.8k
3
votes
Classification of surface bundles over surfaces
About 6 years ago there was an Oberwolfach meeting on surface bundles, and most of the talks were recorded and can be seen here. If I remember correctly, Benson Farb’s overview talk was particularly …
- 44.8k
11
votes
Covers of Riemann surfaces which become arbitrary close in Teichmuller space
At least for the Weil-Petersson metric on Teichmuller space, this is a well-known open problem known as the Ehrenpreis conjecture. It has a rather fearsome reputation.
- 44.8k
7
votes
Index of the mapping class group $\Gamma_{g,n}$ inside $\text{Out}(\Pi_{g,n})$
It only has finite index in very low-complexity degenerate cases.
Here's a proof that it always has infinite index for $\Sigma_{g,1}$ with $g \geq 2$. This proof generalizes in an obvious way to deal …
- 44.8k
4
votes
Accepted
Question related to the moduli space of Riemann surfaces and a fibration
I'll give you references for the appropriate fact about the mapping class group. Let $Mod_{g,b}^p$ be the mapping class group of a genus $g$ surface with $b$ boundary components and $p$ punctures $\S …
- 44.8k
12
votes
Conceptual proof of classification of surfaces?
The proof of Zeeman described in this note is by a substantial margin the easiest and most conceptual proof I know. To simplify the exposition I restrict to orientable surfaces in the note, but it is …
- 44.8k