Research

Let S and X be two connected topological surfaces without boundary, and assume that is either of infinite type or of finite type with negative Euler characteristic. In this paper, we prove that if p: S ---> X is a fully ramified branched covering map, then p satisfies the Birman-Hilden property. This generalizes a theorem of Winarski, and the known results in the literature, to the context of surfaces of infinite type and branched covering maps of infinite degree. We provide many applications.

This work presents the opening chapter of a future book on the symmetries and mapping class groups of finite andinfinite-type surfaces. We systematically develop the theory of topological surfaces, emphasizing the classification of compact and non compact surfaces via their end spaces and orientability types. It serves as a pedagogical reference for students and researchers alike.

In this survey (in Spanish) we give a self-contained and elementary proof that the group of homeomorphisms Homeo(S) of a topological surface S (possibly nonorientable and possible with non-compact boundary components) is a Polish group with compact-open topology. We translate the isotopy relation in Homeo(S) as path-connectedness in Homeo(S). We use classical results from Descriptive Set Theory to prove that the Extended Mapping Class Group Mod(S) of every topological surface is a Polish group with the quotient topology. At the end we discuss an alternative proof of this result which is based on viewing Mod(S) as the automorphism group of the curve graph.

We study the Euler class of smooth orientable infinite-type surface bundles with a section. For many such surfaces, we show that this cohomology class is nontrivial, and that the behavior of its powers depends on the genus and the type of ends. As an application, we extend Morita's non-lifting theorem to many infinite-type surfaces, including surfaces of infinite genus.

We obtain a complete characterization of those that are globally CB, which does not require the tameness condition. We prove that, for surfaces with a unique maximal end, any locally CB big mapping class group is CB generated. Finally, we give an example of a non-tame surface whose mapping class group is CB generated but is not globally CB, answering a question of Mann and Rafi.

In this paper we compute the asymptotic dimension of all surface braid groups, including those associated to non-orientable and infinite-type surfaces, as well as for torsion-free surface groups that are poly-finitely generated. In these two classes of groups we show that the virtual cohomological dimension and the asymptotic dimension coincide.

We show that any countable infinite group can be represented as the full group of automorphisms of a suitable origami on the Loch Ness Monster.

We describe the topological behavior of the conjugacy action of the mapping class group of an orientable infinite-type surface. Our techniques are based on model-theoretic methods developed by Kechris, Rosendal and Truss.

We generalice the Thurston-Veech construction of pseudo-Anosov elements to the setting of infinite-type surfaces. We use it to produces infinitely many loxodromic elements for the action of Mod(S;p) on the loop graph L(S;p) that do not leave any finite-type subsurface invariant.

We associate to each zero-twist flute surface a sequence of positive real numbers. We charactarize when the surface is of first type and when it is of parabolic type in terms of this sequence. The last was afther the work of Basmajian, Hakobian and Šarić. In addition, we present an uncountable family of hyperbolic surfaces homeomorphic to the Loch Ness Monster.

The main contributions of this doctoral thesis are: 1. Alexander's method for infinite-type surfaces. For any infinite-type surface S possibly with non-empty boundary, there exist a collection τ of essential curves and arcs in S such that if h : S → S is a homeomorphism which preserves the isotopy class of the elements in Γ then h is isotopic (relative to the boundary of S) to the identity. 2. Topological rigidity of the curve complex. Every simplicial isomorphism between curve complexes of infinite-type surfaces is induced by a homeomorphism. 3. The automorphisms group of the curve complex. For every infinite-type surface S, the simplicial automorphisms group of the curve complex associated to S is isomorphic to the extended mapping class group of S, that is, Aut (C(S)) = Mod*(S). 4. The Hooper, Thurston and Veech construction. Let S be an infinite-type surface. Then there exist a pair of multicurves α and β whose union fills S, and there exist a flat structure τ on S such that the Dehn twists Tα and Tβ around the multicurves α and β, respectively, are a ne automorphisms of the flat surface M := (S; Γ). 5. Loxodromic big mapping classes. Let S be an infinite-type surface and p be a marked point in S. Then there exists an uncountable collection of elements in the group Mod(S; p) with loxodromic action on the loop graph of S. Moreover, every element of this collection does not preserve any finite-type subsurface of S.

We show that the natural action of the mapping class group of an infinite-type surface on the curve complex is faithful. In particular, we show that there is a countable collection of curves on the surface such that if a homeomorphism fix the isotopy class of each curve in this collection then it is isotopic to the identity.

We show that any simplicial automorphism of the curve graph of an infinite-type surface is induced by a homeomorphism. We also prove that the curve graph of an infinite-type surface is topologically rigid.

Compilation of notes from the 'Lecciones Matemáticas del PCCM'

Given a surface of finite type S, the complex curves C(S) is an abstract simplicial complex associated to the surface S where the vertices are all isotopy class of essential simple closed curves and the k?simplex are collections of k + 1 distint vertices that have pairwise disjoint representatives. The complex curves have been used in many problems in low-dimensional topology and geometry, in the ending laminations conjecture of Thurston for 3-dimensional hyperbolic manifolds and in the quasi-isometric rigidity problem of the modular group Mod(S). In this thesis the main properties of the complex curves as the connexity, the infinite diameter and the uniform hyperbolicity are studied.