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Consider hyperbolic metrics on $\Sigma_g$ a closed orientable surface of genus $g$. Let $[\gamma_1] , \cdots, [\gamma_n]$ be a finite collection of isotopy classes of simple closed curves on $\Sigma_g …

I consider the standard embedding of a compact oriented surface $\Sigma$ (say of genus 2) in the Euclidean space $\mathbb{R}^3$. I have coloured on the picture below the zone of this surface where the …

Let $G$ be a group. It acts canonically on its derived subgroup by conjugation. Can on describe the orbits of this action when $G$ is the fundamental group of a compact orientable surface of genus $g …

I was wondering if anyone knew a 'simple' proof of the fact that a pseudo-Anosov diffeomorphism of a closed surface $\Sigma$ is not reducible, in the sense that it does not fix the free homotopy class …

I was thinking of Veech surfaces, which are translation surfaces whose stabilizer under the $\mathrm{Sl}_2(\mathbb{R})$ action is a lattice in $\mathrm{Sl}_2(\mathbb{R})$. … For example do we know explicit examples of translation surfaces whose Veech group is non-elementary but not a lattice? Thank you for your attention! Selim …

Hello everyone, Let $S_g$ be a compact orientable surface of genus $g \geq 2$, and let $\mathcal{A}$ be the set of $\mathcal{C}^{\infty}$ Riemanniann metric on $S_g$ endowed with the topology of unif …

Let $S$ be a compact orientable surface of genus $g$. Now let $p\in S$ and $\gamma$ a closed simple curve on $S$ disjoint from $p$. It is not very difficult to compute the action of a Dehn twist along …

Let $S$ be a compact translation surface (i.e. a surface endowed with a singular flat metric such that singular points are locally isometric to a cone of angle an integer multiple of $2\pi$, and that …

Let $S$ be a $2$-dimensional sphere endowed with a flat metric with $3$ conical singularities of positive curvature. Typically, $S$ is a metric space you get when you glue two copies of the same trian …

Obvious example of such surfaces are linear planes and the level sets of quadratic form. … Can one list all such surfaces? Immersed or degenerated surfaces would be interesting to me as well. Any reference is welcome! Thanks :) …

I am interested in the following problem : decide if a certain element of the fundamental group can be represented by a simple closed curve. The general case has already been asked and answered on MO …