On the equidistribution of unstable curves for pseudo-Anosov diffeomorphisms of compact surfaces

We prove that the asymptotics of ergodic integrals along an invariant foliation of a toral Anosov diffeomorphism, or of a pseudo-Anosov diffeomorphism on a compact orientable surface of higher genus, is determined (up to a logarithmic error) by the action of the diffeomorphism on the cohomology of the surface. As a consequence of our argument and of the results of Giulietti and Liverani [Parabolic dynamics and anisotropic Banach spaces. J. Eur. Math. Soc. (JEMS)21(9) (2019), 2793–2858] on horospherical averages, toral Anosov diffeomorphisms have no Ruelle resonances in the open interval $(1, e^{h_{\mathrm {top}}})$ .

Reversing orientation homeomorphisms of surfaces

Let $M$ be a connected compact orientable surface, $f:M\to \mathbb{R}$ be a Morse function, and $h:M\to M$ be a diffeomorphism which preserves $f$ in the sense that $f\circ h = f$. We will show that if $h$ leaves invariant each regular component of each level set of $f$ and reverses its orientation, then $h^2$ is isotopic to the identity map of $M$ via $f$-preserving isotopy. This statement can be regarded as a foliated and a homotopy analogue of a well known observation that every reversing orientation orthogonal isomorphism of a plane has order $2$, i.e. a mirror symmetry with respect to some line. The obtained results hold in fact for a larger class of maps with isolated singularities from compact orientable surfaces to the real line and the circle.

The mapping class group action on -character varieties

Let $\unicode[STIX]{x1D6F4}$ be a compact orientable surface of genus $g=1$ with $n=1$ boundary component. The mapping class group $\unicode[STIX]{x1D6E4}$ of $\unicode[STIX]{x1D6F4}$ acts on the $\mathsf{SU}(3)$ -character variety of $\unicode[STIX]{x1D6F4}$ . We show that the action is ergodic with respect to the natural symplectic measure on the character variety.

Algorithm for filling curves on surfaces

Let $$\varSigma $$ Σ be a compact, orientable surface of negative Euler characteristic, and let h be a complete hyperbolic metric on $$\varSigma $$ Σ . A geodesic curve $$\gamma $$ γ in $$\varSigma $$ Σ is filling if it cuts the surface into topological disks and annuli. We propose an efficient algorithm for deciding whether a geodesic curve, represented as a word in some generators of $$\pi _1(\varSigma )$$ π 1 ( Σ ) , is filling. In the process, we find an explicit bound for the combinatorial length of a curve given by its Dehn–Thurston coordinate, in terms of the hyperbolic length. This gives us an efficient method for producing a collection which is guaranteed to contain all words corresponding to simple geodesics of bounded hyperbolic length.

Mapping Class Groups

This chapter considers the mapping class group, the group of symmetries of a surface, and some of its basic properties. It first provides an overview of surfaces and the concept of homeomorphism before giving examples of homeomorphisms and defining the mapping class group as a certain quotient of the group of homeomorphisms of a surface. It then looks at Dehn twists and describes some of the relations they satisfy. It also presents a theorem stating that the mapping class group of a compact orientable surface is generated by Dehn twists and proves it. It concludes with some projects and open problems. The discussion also includes various exercises.

On the Connectedness of Moduli Spaces of Flat Connections over Compact Surfaces

We study the connectedness of the moduli space of gauge equivalence classes of flat G-connections on a compact orientable surface or a compact nonorientable surface for a class of compact connected Lie groups. This class includes all the compact, connected, simply connected Lie groups, and some non-semisimple classical groups.

Cyclic p-groups of symmetries of surfaces

Let Σg denote a compact orientable surface of genus g ≥ 2. We consider finite groups G acting effectively on Σg and preserving the orientation—for short, G acts on Σg or Gis a symmetry group of Σg. Each surface Σg admits only finitely many symmetry groups G and the orders of these groups are bounded by Wiman's bound of 84(g – 1). This bound is attained for infinitely many values of g [12], see also [9], and all values of g ≤ 104 for which it is attained are known [4].