Research

Research interests

Classical and big mapping class groups, groups actions on simplicial complexes, Teichmüller theory, flat surfaces, hyperbolic geometry.

Preprints

• On the large scale geometry of big mapping class groups of surfaces with a unique maximal end. With Rita Jiménez-Rolland. arXiv:2309.05820. Submitted (2023). Acepted for publication in Michigan Math. J.

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.

Papers

1. Isomorphisms between curve graphs of infinite-type surfaces are geometric. With Jesús Hernández Hernández and Ferrán Valdez.  Rocky Mountain J. Math. 48(6): 1887-1904 (2018). DOI: 10.1216/RMJ-2018-48-6-1887. 

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.

2. The Alexander method for infinite-type surfaces. With Jesús Hernández Hernández and Ferrán Valdez. Michigan Math. J. 68(4): 743-753 (November 2019). DOI: 10.1307/mmj/1561773633. 

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. 

3. Loxodromic elements in big mapping class groups via the Hooper-Thurston-Veech construction. With Ferrán Valdez. Algebraic & Geometric Topology 22 (2022) 3809–3854. DOI: 10.2140/agt.2022.22.3809. 

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.

4. Conjugacy classes of big mapping class groups. With Jesús Hernández Hernández,  Michael Hrusák,  Anja Randecker, Manuel Sedano and Ferran Valdez. J. London Math. Soc., 106: 1131-1169. https://doi.org/10.1112/jlms.12594. 

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.

5. Parabolicity of zero-twist tight flute surfaces and uniformization of the Loch Ness monster. With John A. Arredondo & Camilo Ramírez Maluendas. Lobachevskii J. Math 43, 10–20 (2022). https://doi.org/10.1134/S1995080222040035.

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.

6. Realising countable groups as automorphisms of origamis on the Loch Ness monster. With Rubén Hidalgo. Arch. Math. 120, 355–360 (2023). https://doi.org/10.1007/s00013-023-01835-4. 

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. 

Other writings (spanish)

• Acciones simpliciales de grupos modulares de superficies de tipo infinito. PhD thesis (2020).

• Hiperbolicidad Uniforme del complejo de curvas. Master's thesis (2016).

• Memorias Matemáticas del PCCM. With Héctor A. Barriga Acosta (editors). Compilation of notes from the "Lecciones Matemáticas del PCCM" (2017).