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Research

Preprints

  1. Rigidity of singular de-Sitter tori with respect to their lightlike bi-foliation (2024). Submitted.
    arXiv:2410.03260, hal-04661986

    In this paper, we introduce a natural notion of constant curvature Lorentzian surfaces with conical singularities, and provide a large class of examples of such structures. We moreover initiate the study of their global rigidity, by proving that de-Sitter tori with a single singularity of a fixed angle are determined by the topological equivalence class of their lightlike bi-foliation. While this result is reminiscent of Troyanov's work on Riemannian surfaces with conical singularities, the rigidity will come from topological dynamics in the Lorentzian case.

  2. Reductions of path structures and classification of homogeneous structures in dimension three (with Elisha Falbel and Jose Miguel Veloso, 2024). Submitted.
    arXiv:2406.11509, hal-04612815

    In this paper we show that if a path structure has non-vanishing curvature at a point then it has a canonical reduction to a $\mathbb{Z}/2\mathbb{Z}$-structure at a neighbourhood of that point. A simple implication of this result is that the automorphism group of a non-flat path structure is of maximal dimension three (a result by Tresse of 1896). We also classify the invariant path structures on three-dimensional Lie groups.

  3. Geometric surgeries of three-dimensional flag structures and non-uniformizable examples (with Elisha Falbel, 2024). Submitted.
    arXiv:2406.02053, hal-04598551

    In this paper, we introduce a notion of geometric surgery for flag structures, which are geometric structures locally modelled on the three-dimensional flag space under the action of $\mathrm{PGL}_3(\mathbb{R})$. Using such surgeries we provide examples of flag structures, of both uniformizable and non-uniformizable type.

Publications

  1. Geometrical compactifications of geodesic flows and path structures.
    Geometriae Dedicata 217(2) (2022), 18.
    arXiv:2112.02900, hal-03466455

    In this paper, we construct a geometrical compactification of the geodesic flow of non-compact hyperbolic surfaces $\Sigma$ without cusps having a finitely generated fundamental group, and we study the dynamical properties of the compactified flow. This compactification is done with respect to a uniformizable flag structure, for which ${\mathrm{T}}^1\Sigma$ is identified to the quotient of an open subset of the flag space by a discrete subgroup $\Gamma$ of ${\mathrm{PGL}}_3(\mathbb{R})$. Our study relies on a detailed description of the dynamics of ${\mathrm{PGL}}_3(\mathbb{R})$ on the flag space, and on the construction of an explicit fundamental domain for the action of $\Gamma$.

  2. Cartan connections and path structures with large automorphism groups (with Elisha Falbel and Jose Miguel Veloso).
    International Journal of Mathematics 32(12) (2021).
    arXiv:2105.02090, hal-03214060

    We classify compact manifolds of dimension three equipped with a path structure and a fixed contact form (which we refer to as a strict path structure) under the hypothesis that their automorphism group is non-compact. We use a Cartan connection associated to the structure and show that its curvature is constant.

  3. Partially hyperbolic diffeomorphisms and Lagrangian contact structures.
    Ergodic Theory and Dynamical Systems 42(8) (2022), 2583-2629.
    arXiv:2002.10720, hal-02489310

    In this paper, we classify the three dimensional partially hyperbolic diffeomorphisms whose invariant distributions are smooth, such that $E^s\oplus E^u$ is a contact distribution and whose non-wandering set equals the whole manifold. We prove that up to a finite quotient or a finite power, they are smoothly conjugated either to the time-one map of an algebraic contact-Anosov flow, or to an affine partially hyperbolic automorphism of a nil-manifold. The rigid geometric structure induced by the invariant distributions plays a fundamental role in the proof.

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