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How exactly do we put hyperbolic structures on a sphere with cone point singularities. Should I consider that sphere with cone points as an extended complex plane with punctures endowed with a suitab …

Given a closed surface of genus $g\geq 2$ and a fixed hyperbolic metric on it, how many pants decompositions exist for that surface? I tend to believe that it is finite ? For example, if we take a s …

I googled it and wikipidead it too : but apparently there is no definition of an ideal triangle on a punctured torus ( i.e a compact [ hyperbolic ] surface with one genus and one boundary component, o …

Consider a figure-eight geodesic $\delta $( geodesic with exactly one self-intersection point at p ) on a pair of pants Y with three geodesic boundaries $ \gamma_i$ and three perpendiculars between th …

This is a basic question, still I dare to ask : Let Y be the Y-piece with geodesic boundaries A,B, C and ( if possible ) c the non simple geodesic from A to B intersecting itself at a point p. I want …

I was trying to prove that $ Aut( S_g $), g$ \geq 2 $ [ orientation preserving isometries ] is finite in the following way : For fixed $M $ ( positive ) there are finitely many , say $ k $ number of …

I know the definition of k -quasi-symmetric maps $f$ on $R$,it is there exists $k>0$ such that $\frac{1}{k}\leq\frac{f(x+t)-f(x)}{f(x)-f(x-t)} \leq k \forall x,t\in R.$ So I just want to double c …

Hello, as we know, the (finite) Blaschke product $P$ in $\mathbb{C}$ or in $ \mathbb{R}^2 $ is defined by $\prod_{j=1}^{k} \frac{z-a_j}{1-\bar{a_j}z}, a_j \in \mathbb{D}$. I was wondering whether the …

Hello, Is there any general ways to construct compact hyperbolic 2-manifolds with very small or very large eigenvalues ? Also, as a special case, can we construct a sequence of compact hyperbolic 2- …

Hello, Could you name a couple of books or downloadable lecture notes that discuss spectral graph theory and its connection to spectral problems in hyperbolic Riemann surfaces ? You could also mentio …

Let us take two copies of $ Y $-pieces [ or pair of pants ] with each boundary geodesic of length $ l $, and glue them together without any twisting to obtain a genus 2 closed orientable hyperbolic s …

I know that this definitely have some sort of reference out there, but I did not find any wikipidea page for it or any introductory Mathematical article about it . I just want definition and concrete …

If we take a sequence of compact hyperbolic Riemann surface with k geodesic boundary components such that the lengths of the geodesic boundary components go to zero, then in the "limit", we should get …

I was studying the proof of Teichmuller's uniqueness theorem from the note/book " A Primer on Mapping Class Groups " by Farb-Margalit and I got struck at a couple of points, mainly because I am new to …

Let $f,g : X \to Y$ be homotopic (quasiconformal) maps between hyperbolic Riemann surfaces $X,Y$. Consider their (unique) lifts $\tilde{f},\tilde{g}: H\to H$ , that fix $0,1,\infty $. My question is : …