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Let $S$ be a hyperbolic surface. Suppose $\mathcal{T}$ denotes the Teichmuller space of $S$ and $Mod(S)$ denotes the mapping class group of $S$. Given any pseudo-Anosov element $f\in Mod(S)$, suppose …

In Fifteen problems about MCG Ivanov stated the following metaconjecture: Every object naturally associated to a surface S and having a sufficiently rich structure has $Mod(S)$ as its groups of autom …

Let $S$ be a closed hyperbolic surface. Suppose $Mod(S)$ denotes the mapping class groups and $T(F)$ denotes the Teichmüller space. By Birman exact sequence we get the point pushing map $Push:\pi_1(S …

In the paper "Triangulations of Surfaces" Hatcher proved that the arc complex associated to a punctured surface is contractible. The main proof is divided into two parts. In the first part he assumes …

Can someone please give any link or mention any source where I can find the following preprint. W.Thurston, A spine for Teichmüller space, preprint, three pages, 1986.

Let $S$ be a surface of negative Euler characteristic. Consider the Birman exact sequence: $$1\xrightarrow{ }\pi_1(S,p)\xrightarrow{P} Mod(S,p)\xrightarrow{ }Mod(S)\xrightarrow{ }1$$ In his paper he …

Given a closed surface of genus $g\geq 2$, we know that the mapping class group $Mod(S)$ is generated by the Dehn twists. My question is Given an element as a product of Dehn twist, is it possible …

Let $F$ be an oriented surface of finite type with $\chi(F)<0$. Let $\gamma_1$ and $\gamma_2$ are two oriented closed curves which intersect transversally in double points. Given a hyperbolic metric i …

Let $F$ be a hyperbolic surface and $p\in F$ be a point. Consider $\pi_1(F,p)$, the fundamental group of $F$ with base point $p$. Let $x,y\in \pi_1(F,p)$ and $z$ be a simple closed curve in $F$ such t …

Let $S$ be a fixed hyperbolic surface with genus $g$ and $n$ punctures. Given any pseudo-Anosov map $f$ on $S$ (with stretch factor $\lambda$) with stable and unstable measured foliations $\mu^s$ and …

In the paper "The Nielsen realization problem" (here), Kerckhoff proved that the length function on the Teichmüller for closed surface is convex. In his paper "Geodesic length functions and the Nielse …

Let $S$ be a closed hyperbolic surface and $x$ be an oriented simple closed curve in $S$. Let $y$ be an oriented closed curve such that the geometric intersection number between $x$ and $y$ is positi …

There are various complexes associated to a surface using the curves and arcs e.g. Curve complex, Arc complex, curve arc complex and so on (for a collection of such objects see This). Now to understa …

Let $S$ be a hyperbolic surface of finite type and $\alpha,\beta$ be two closed curves. Consider $X$ to be the set of all those points $\chi$ in the Teichmuller space $\mathcal{T}(S)$ of $S$ such that …

Let $S$ be a closed oriented surface of genus $g\geq 2$. Consider the Teichmuller space $T(S)$. Let $d_t$ be the Teichmuller metric and $d_{WP}$ be the Weil-Petersson metric on $T(S)$. Let $P:S_1\righ …