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Mirzakhani proved identities for the lengths of geodesic curves on Riemann surfaces of genus $g$ and with $n$ boundary components. She used these to provide an integration scheme over the correspondin …

I will base my question on Fock and Goncharov's paper Dual Teichmüller and lamination spaces. Let $S$ be a surface with boundaries, marked points on such boundaries, punctures and boundaries without m …

I would like to understand, as explicitly as possible, how different coordinates on the moduli space of Riemann surfaces are related: On the one hand, there is a parametrization coming from hyperbolic …

Let $S$ be an orientable surface with a triangulation T. A lamination $\ell$ is a simple closed curve on $S$, up to isotopy. We will assume that $\ell$ is drawn in such a way that it intersects the ed …

Using Fenchel-Nielsen coordinates, the Weil-Petersson metric can be written as $\omega_{WP} = \sum_{i} d\ell_i \wedge d \tau_i,$ where $i$ is an index labelling the curves of a pants decomposition of …

In the complex analytic setting, it is easy to see that the moduli space of a sphere with four punctures is $\mathcal{M}=\mathbb{CP}^1 / { 0,1,\infty }$, since I can use a Moebius transformation to se …

The decorated Teichmuller space of a disk with n punctures on the boundary and one in the interior is the the space of hyperbolic metrics on such a surface with an extra marking of an horocycle at eac …

Certain cluster algebras arise from ideal triangulations of hyperbolic Riemann surfaces. The combinatorics behind their mutations can be understood in terms of "flips" in the triangulation, and the cl …

In (https://www.sciencedirect.com/science/article/pii/002240499390049Y) it is mentioned the real section of the moduli space of Riemann surfaces of genus 0. It can be intuitively defined as a subset w …

Any annulus is biholomorphich to the Poincare' disk $D$ from wich a circle centered at the origin has been removed. If we want to parametrise annuli with punctures at one boundary, give the punctures …

The automorphism group of the Poincare' disk has elements called elliptic, which have a single fixed point in the interior of the disk, and can be represented as a rotation around this fixed point. I …

Consider the moduli space of hyperbolic metrics on the disk with $n>3$ marked points on its boundary, $\mathcal{M}_{D,n}$. $\mathcal{M}_{D,n}$ can be parametrised in terms of cross ratios of the punc …

Let D be the Poincare' disk its natural hyperbolic metric and with at least 1 marked point on $\partial D$. Suppose I cut an hyperbolic circle of radius $r$ away from it, then I get a Riemann surface …

I already asked this question on math.stackexchange, but it was suggested that I post it here as well. The paper Devadoss, Heath, and Vipismakul - Deformations of bordered Riemann surfaces and associ …