6
votes
0answers
112 views

Is there an analogue of Mostow-Palais equivariant embedding theorem for noncompact groups

Let $M$ be a (Hausdorff) smooth compact manifold and $G$ a Lie group acting smoothly on $M$. If $G$ is compact then, by Mostow-Palais theorem, there exists an equivariant smooth embedding $M\to ...
5
votes
1answer
285 views

Universal covering map from $\mathcal{H}$ to $\mathbb{C}\setminus \mathbb{Z}\oplus i\mathbb{Z}$ (the countably punctured complex plane)

It's a consequence of the uniformization theorem for simply connected Riemann surfaces that the universal cover of $\mathbb{C}\setminus(\mathbb{Z}\oplus i\mathbb{Z})$ ($\mathbb{C}$ punctured at all ...
16
votes
2answers
375 views

A direct proof of the Harer-Zagier recursion enumerating the ways to paste a 2n-gon to get a genus g surface?

In a 1986 paper, Harer and Zagier proved the recursion: $$(n+1)e(g,n)=(4n-2)e(g,n-1)+(2n-1)(n-1)(2n-3)e(g-1,n-2)$$ where e(g,n) is the number of ways of grouping sides $S_1...S_{2n}$ of a 2n-gon ...
10
votes
4answers
2k views

Geometric invariant theory for geometers

I am trying to learn "Geometric invariant theory" like it was introduced by Mumford. But I do not have a strong background in algebraic geometry since I work in geometric topology and geometry. So ...
3
votes
3answers
363 views

Jordan curve theorem for cylinders

Hello, I would like to know if the following result is true: Let $A,B$ be two embedded circles in $S^2$ which do not intersect and let $C$ be the $\textit{closed}$ region bounded by $A$ and $B$ ...
1
vote
1answer
165 views

Quasi-conformal map between Riemann surfaces with punctures

Let $S$ be a Riemann surface of genus $g \geq 0$ with $n$ punctures, i.e., with $n$ distict points removed. Let $f: S \rightarrow R$ be a quasi-conformal map. Then $R$ is also Riemann surface of genus ...
2
votes
1answer
546 views

How to calculate Dr. Curt McMullen's expanding eigenvalues for totally degenerate groups?

What is required in order to derive the expanding eigenvalues of Dr. Curt McMullen's torus orbifold bundles over the circle and the corresponding totally degenerate groups, as presented in Section 3.7 ...
18
votes
2answers
1k views

Riemann mapping theorem for homeomorphisms

How do you prove to any two simply-connected domains in the plane are homeomorphic without using the Riemann mapping theorem? An elementary proof would be nice.
1
vote
0answers
138 views

Generalized vector bundles with singularities on Riemann surfaces

Let $X$ be a Riemann surface of genus $g \geq 2$ or in other words a complex curve. Let $P_1, \ldots, P_m$ be points in $X$ and $E \rightarrow X$ surjective map such that is is a complex ...
5
votes
0answers
494 views

Computing the Chern class for a flat line bundle using the holonomy group?

Let $X$ be a Riemann surface of genus $g \geq 2$. I would like to consider flat line bundles on $X$. Flat line bundles can be identified with representations $\pi_1(X) \rightarrow U(1)$. From ...
1
vote
2answers
274 views

Families of Fuchsian models

A Fuchsian model for a Riemann surface $X$ is a discrete subgroup $G$ of $PSL_2(\mathbb{R})$ such that there is a biholomorphic map from $U/G$ to $X$. For a fixed genus $g \geq 2$ one knows from Bers ...
5
votes
2answers
385 views

Surface automorphisms and conformal automorphisms

Given a closed orientable surface $S$ and a topological automorphism $\sigma$ of $S$, it is not in general possible to find a conformal structure $\Sigma$ on $S$ so that $\sigma$ is isotopic to a ...
9
votes
1answer
490 views

on common fixed points of commuting polynomials (and rational functions)

By the Ritt's classification, for any pair of commuting polynomials (i.e. $f(g(z))=g(f(z))$) over $\mathbb C$ there is a common fixed point of them. My questions are: Is that true that this can be ...
4
votes
1answer
385 views

monodromy of plane curve singularities

Are there two IRREDUCIBLE plane curve singularities having different equisingular type with the same monodromy (linear action on the first homology group of the (regular) Milnor fibre)?
4
votes
3answers
690 views

How can one express the Dedekind eta function as a sum over the lattice?

The Dedekind eta function $\eta(\tau)$ can be regarded as a formula which assigns a number to a lattice $L \subset \mathbb{C}$. The algorithm is: rotate the lattice so that one of its basis vectors ...
93
votes
1answer
8k views

What are the shapes of rational functions?

I would like to understand and compute the shapes of rational functions, that is, holomorphic maps of the Riemann sphere to itself, or equivalently, ratios of two polynomials, up to Moebius ...
6
votes
1answer
634 views

Simply-connected domain around a curve

In a current project with a colleague, we have come across the following reasonably classical-sounding geometric question. While not vital to our work, it would be interesting if anyone has seen this ...