Research

I am interested in low dimensional topology. In particular I have been studying hyperbolic low dimensional manifolds with non-finitely generated fundamental group. I am also interested in higher Teichmüller Theory and counting problems on surfaces.

Papers:

  1. A locally hyperbolic 3-manifold that is not hyperbolic, Proc. Amer. Math. Soc. 146 (2018), 5475-5483. In this paper we answer a question of Agol by showing that there exists a locally hyperbolic 3-manifolds with no locally divisible subgroups that is not hyperbolic.
  2. Discrete groups without finite quotients with Juan Souto, Topology and its Applications, 2018, Vol.248, pp.138-142. In this paper we build an exotic example of a non-trivial sub-group of isometry of hyperbolic 3-space that has no non-trivial quotients. This corresponds to a hyperbolic 3-orbifold. We also construct a hyperbolic 3-manifold that is not residually finite.
  3. Hyperbolicity of links complements in Seifert fibered spaces with José Andrés Rodríguez Migueles, Algebraic and Geometry Topology in 2020.  In this paper we give upper bounds for the volume of knots in Seifert-fibered spaces based on the complexity of their projection on their Seifert-surface.
  4. A locally hyperbolic 3-manifold that is not homotopy equivalent to any
    hyperbolic 3-manifold, Conform. Geom. Dyn. 24 (2020), 118-130. In this paper we improve the example in our first paper by showing that there exists a locally hyperbolic 3-manifolds with no locally divisible subgroups that is not homotopy equivalent to any hyperbolic 3-manifold.
  5. Hyperbolic limits of Cantor set complements in the sphere with Franco Vargas Pallete; Bulletin of the London Mathematical Society, vol. 54, issue 3, pages 1104-1119,  2022. In this paper we show that hyperbolic Cantor set complements in the 3-sphere are dense, with respect to the geometric topology, in the set of hyperbolic manifolds without rank two cusps that admit embeddings in the 3-sphere.
  6. On volumes and filling collections of multicurves with José Andrés Rodríguez Migueles and Andrew Yarmola;  Journal of Topology, 15(3):1107–1153, 2022. In this paper we study links in the projective bundle of surfaces that arise as canonical lifts of filling collections of simple closed curves. For large classes of these links we give volumes asymptotics that involve combinatorial data coming the curves.
  7. Effective contraction of skinning maps, joint work with Lorenzo Dello Schiavo, accepted at Proceedings of the AMS, 2022. In this work we give explicit bounds on the contraction factor of the skinning map over the moduli space of hyperbolic surfaces.
  8. Volume bounds for the canonical lift complement of a random geodesic, joint work Yannick Krifka, Didac Martinez-Granado , Franco Vargas Pallete; accepted at Transactions of the AMS. In this paper we study random geodesic obtained by flowing in random directions under the geodesic flow and their canonical lifts in the projective tangent bundle. For this random model we give a probabilistic estimate of the volume of the corresponding lift complement that is sub-linear with respect to the length of the geodesic. 

Pre-Prints: 

    1. Hyperbolization of infinite type 3-manifolds; April 2019 pre-print, submitted. In this paper we give the first known hyperbolisation results for a large class of infinite-type hyperbolic 3-manifolds.
    2. Fibered 3-manifolds with unique incompressible surfaces; joint with Andrew Yarmola, August 2023 pre-print, submitted. In this short work we build infinitely many closed fibered 3-manifolds that are hyperbolic and have a unique incompressible surface corresponding to the fiber.
    3. Knots in circle bundles are determined by their complement; joint with Andrew Yarmola, January 2024. We resolve a case of the oriented knot complement conjecture by showing that knots in an orientable circle bundle N over a genus g≥2 surface S are determined by their complements. 
    4. Density results for the modular group of infinite-type surfaces; joint with Yassin Chandran, March 2024. We prove two results about approximating, with respect to the compact-open topology, mapping classes on surfaces of infinite-type by quasi-conformal maps, in particular we are interested in density results, i.e. when the modular group is dense in the mapping class group. We show that his holds for any surface with countable end space and for any surface if we consider the pure mapping class group.

Collaborators: