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;

    Transactions of the American Mathematical Society, Series B 10.28 (2023): 988-1038. 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. 

  9. Fibered 3-manifolds with unique incompressible surfaces; joint with Andrew Yarmola, Journal of Knot Theory and Its Ramifications Vol. 33, No. 10, 2450029 (2024). 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.
  10. Filling Riemann surfaces by hyperbolic Schottky manifolds of negative volume; joint with Viola Giovanni and Jean-Marc Schlenker. In this work we provide conditions under which a Riemann surface X is the asymptotic boundary of a convex co-compact hyperbolic manifold, homeomorphic to a handlebody, of negative renormalized volume. We prove that this is the case when there are on X enough closed curves of short enough hyperbolic length. Accepted at Journal of Geometry and Physics, summer 2025.
  11. Covers of Surfaces; joint with Ian Biringer, Yassin Chandran, Jing Tao, Nicholas G. Vlamis, Mujie Wang, Brandis Whitfield . We study the homeomorphism types of certain covers of (always orientable) surfaces, usually of infinite-type. We show that every surface with non-abelian fundamental group is covered by every noncompact surface, we identify the universal abelian covers and the ℤ/nℤ-homology covers of surfaces, and we show that non-locally finite characteristic covers of surfaces have four possible homeomorphism types. Accepted at AGT summer 2025.

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. 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. 
    3. 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.
    4. Monomial web basis for the SL(N) skein algebra of the twice punctured sphere; joint with Daniel Douglas. In this work we show that for any non-zero complex number q, excluding finitely many roots of unity of small order, a linear basis for the SL(n) skein algebra of the twice punctured sphere is constructed. In particular, the skein algebra is a commutative polynomial algebra in n−1 generators, where each generator is represented by an explicit SL(n) web, without crossings, on the surface. This includes the case q=1, where the skein algebra is identified with the coordinate ring of the SL(n) character variety of the twice punctured sphere. The proof of both the spanning and linear independence properties of the basis depends on the so-called SL(n) quantum trace map, due originally to Bonahon-Wong in the case n=2. A consequence of the proof is that the polynomial algebra sits as a distinguished subalgebra of the Lê-Sikora SL(n) stated skein algebra of the annulus. We end by discussing the relationship with Fock-Goncharov duality.

    5. Behaviour of the Schwarzian derivative on long complex projective tubes; joint with Viola Giovannini, February 2025.  In this work we study the behaviour of the Scharzian derivative along short geodesics. The Schwarzian derivative parametrizes the fibres of the space of complex projective structures on a surface as vector bundle over its Teichmüller space. We study its behaviour on long complex projective tubes, and get estimates for the pairing of its real part with infinitesimal earthquakes and graftings. As the real part of their Schwarzian coincides with the differential of the renormalized volume we obtain bounds for the variation of renormalized volume under complex earthquake paths, and its asymptotic behaviour under pinching a compressible curve.

Collaborators: