Tom Ferragut

Postdoctorant in Geometric group theory

Scientific communication: including talks, (pre)-publications and slides when available.

Publications

  • November 2022: Geometric rigidity of quasi-isometries in horospherical products by T. Ferragut. [Submitted] Abstract: We prove that quasi-isometries of horospherical products of hyperbolic spaces are geometrically rigid in the sense that they are uniformly close to product maps, this is a generalisation of the result obtained by Eskin, Fisher and Whyte in [7]. Our work covers the case of solvable Lie groups of the form R % (N1 × N2), where N1 and N2 are nilpotent Lie groups, and where the action on R contracts the metric on N1 while extending it on N2. We obtain new quasi-isometric invariants and classifications for these spaces.

    [ArXiv]

  • September 2020: Geodesics and Visual boundary of Horospherical Products by T. Ferragut. [Submitted] Abstract: Horospherical products of two hyperbolic spaces unify the construction of metric spaces such as the Diestel-Leader graphs, the SOL geometry or the treebolic spaces. Given two proper, geodesically complete, Gromov hyperbolic, Busemann spaces Hp and Hq, we study the geometry of their horospherical product H:=Hp⋈Hq through a description of its geodesics. Specifically we introduce a large family of distances on Hp⋈Hq. We show that all these distances produce the same large scale geometry. This description allows us to depict the shape of geodesic segments and geodesic lines. The understanding of the geodesics' behaviour leads us to the characterization of the visual boundary of the horospherical products. Our results are based on metric estimates on paths avoiding horospheres in a Gromov hyperbolic space.

    [ArXiv]

  • May 2018: Elementary solution of an infinite sequence of instances of the Hurwitz problem by T. Ferragut, C. Petronio. Rendiconti Lincei Matematica E Applicazioni 29(2):297-307 Abstract: We prove that there exists no branched cover from the torus to the sphere with degree 3h and 3 branching points in the target with local degrees (3,...,3), (3,...,3), (4,2,3,...,3) at their preimages. The result was already established by Izmestiev, Kusner, Rote, Springborn, and Sullivan, using geometric techniques, and by Corvaja and Zannier with a more algebraic approach, whereas our proof is topological and completely elementary: besides the definitions, it only uses the fact that on the torus a simple closed curve can only be trivial (in homology, or equivalently bounding a disc, or equivalently separating) or non-trivial.

    [ArXiv]

Phd:

June 2022: Geometry of horospherial products. by T. Ferragut. [PhD manuscript] Abstract: In this manuscript we study the geometry of some metric spaces called horospherical product. They are constructed out of two Gromov hyperbolic spaces, and contains both discrete or continuous examples such as the Diestel-Leader graphs, the SOL geometry or the treebolic spaces. In the first part of this manuscript, we consider two proper, geodesically complete, Gromov hyperbolic, Busemann spaces X and Y . We construct their horospherical product X ⋈ Y and, after some metric estimations on specific paths in Gromov hyperbolic spaces, we describe a family of distances on X ⋈ Y . More specifically, we show that all these distances produce the same large scale geometry for X ⋈ Y . This description allows us to depict the shape of geodesic segments and geodesic lines. The understanding of the geodesics’ behaviour leads us to the characterization of the visual boundary of X ⋈ Y . For the second part, the two spaces X and Y are endowed with measures. Thanks to these measures, we manage to achieve the geometric rigidity of self quasi-isometries of X ⋈ Y . More specifically, we show that every self quasi-isometry of X ⋈ Y is close to a product map. To do so, we first develop several metric and measure theoretic tools regarding a specific family of geodesic called vertical geodesics. These tools include the coarse differentiation, introduced by Eskin, Fisher and Whyte for the horospherical product of regular infinite trees and hyperbolic planes. Afterwards, generalising techniques they presented, we obtain geometric rigidity. In the last chapter we present an example on how to use this geometric rigidity on X ⋈ Y in order to get informations on its quasi-isometry group. More precisely, we provide a description of the quasi-isometry group of a family of solvable Lie groups.

[Pdf]

Talks

  • February 2023: Quasi-isometry classification of solvable Lie groups as Carnot type horospherical products. at Séminaire groupes et analyse, Université de Neuchâtel. Abstract: Horospherical products of two Gromov hyperbolic spaces where introduced to unify the construction of metric spaces such as Diestel-Leader graphs, the Sol geometry or treebolic spaces. In this talk we will first present the bases required to construct these horospherical products. Then, we will get interested in a geometric rigidity property of their quasi-isometries. Using this geometric rigidity in combination with Pansu derivative, we will show that in the context of Carnot groups, context that we will introduce during the talk, we can achieve a quasi-isometry classification for Carnot type horospherical products.
  • January 2023: Rigidity of quasi-isometries and geometry of horospherical products. at Geometric Group Seminar, University of Münster. Abstract: Horospherical products of two Gromov hyperbolic spaces where introduced to unify the construction of metric spaces such as Diestel-Leader graphs, the Sol geometry or treebolic spaces. In this talk we will first recall all the bases required to construct these horospherical products, then we will study their large scale geometry through a descritption of their geodesics and visual boundary. Afterwards we will get interested in a geometric rigidity property of their quasi-isometries. This result will lead us to a description of the quasi-isometry group of solvable Lie groups constructed as horospherical products and to a new quasi-isometry classification for some solvable Lie groups.
  • December 2022: Geometry and rigidity of quasi-isometries of horospherical products. at Topics in Geometric Analysis II (Fall2022), EPFL. Abstract: Horospherical products of two Gromov hyperbolic spaces where introduced to unify the construction of metric spaces such as Diestel-Leader graphs, the Sol geometry or treebolic spaces. In this talk we will first recall all the bases required to construct these horospherical products, then we will study their large scale geometry through a descritption of their geodesics and visual boundary. Afterwards we will get interested in a geometric rigidity property of their quasi-isometries. This result will lead us to a description of the quasi-isometry group of solvable Lie groups constructed as horospherical products and to a new quasi-isometry classification for some solvable Lie groups.
  • December 2022: Geometry and rigidity of quasi-isometries of horospherical products. at Special Seminar, University of Fribourg. Abstract: Horospherical products of two Gromov hyperbolic spaces where introduced to unify the construction of metric spaces such as Diestel-Leader graphs, the Sol geometry or treebolic spaces. In this talk we will first recall all the bases required to construct these horospherical products, then we will study their large scale geometry through a descritption of their geodesics and visual boundary. Afterwards we will get interested in a geometric rigidity property of their quasi-isometries. This result will lead us to a description of the quasi-isometry group of solvable Lie groups constructed as horospherical products and to a new quasi-isometry classification for some solvable Lie groups.
  • October 2022: Geometry and quasi-isometry rigidity of horospherical products. at Differential Topology Seminar, Kyoto University. Abstract: Horospherical products of two Gromov hyperbolic spaces where introduced to unify the construction of metric spaces such as Diestel-Leader graphs, the Sol geometry or treebolic spaces. In this talk we will first recall all the bases required to construct these horospherical products. Then we will study their large scale geometry through a descritption of their geodesics and visual boundary. Afterwards we will get interested in a geometric rigidity property of their quasi-isometries. This result will lead us to a description of the quasi-isometry group of solvable Lie groups constructed as horospherical products.

    [slides]

  • May 2022: Geometry of horospherical products. at Rencontres mathématiques à Sète. Géométrie, topologie et dynamique en basses dimensions. Domaine du Lazaret, Sète. Abstract: Horospherical products of two Gromov hyperbolic spaces where introduced to unify the construction of metric spaces such as Diestel-Leader graphs, the Sol geometry or treebolic spaces. These examples, coming either from geometric group theory or from the study of solvable Lie groups, share similar rigidity properties. In this talk we will first recall all the bases required to construct these horospherical products. Then we will study their large scale geometry through a geometric rigidity property of their self quasi-isometries. This result will lead us to a description of the quasi-isometry group of solvable Lie groups constructed as horospherical products.
  • April 2022: Quasi-isometry rigidity of horospherical products and quasi-isometry group of some solvable Lie groups. at Geometry and Topology Seminar, Bowling Green University, Ohio. Abstract: Horospherical products of two Gromov hyperbolic spaces where introduced to unify the construction of metric spaces such as Diestel-Leader graphs, the Sol geometry or treebolic spaces. These examples, coming either from geometric group theory or from the study of solvable Lie groups, share similar rigidity properties. In this talk we will first recall all the bases required to construct these horospherical products. Then we will study their large scale geometry through a geometric rigidity property of their self quasi-isometries. This result will lead us to a description of the quasi-isometry group of solvable Lie groups constructed as horospherical products.

    [slides]

  • November 2021: Geodesics and Visual Boundary of Horospherical products at Young researchers meeting of the GDR Platon, CIRM, Luminy, Marseille. Abstract: FILS IN
  • January 2021: Geodesics and Visual Boundary of Horospherical products at Strukturtheorie Seminar, [Online], Institut für Diskrete Mathematik, Graz. Abstract: FILS IN
  • March 2020: Geodesics and Visual Boundary of Horospherical products at Séminaire Darboux, [Online], University of Montpellier. Abstract: FILS IN
  • March 2020: Horospherical products of Gromov hyperbolic spaces at Dynamics and Group Geometry Early Researchers Seminar (DAGGER), University of Warwick. Abstract: FILS IN
  • March 2020: Horospherical products of Gromov hyperbolic spaces at Bristol Junior Geometry Seminar (BRIJGES), University of Bristol. Abstract: FILS IN
  • March 2020: Horospherical products of Gromov hyperbolic spaces at Junior Topology and Group Theory Seminar, University of Oxford. Abstract: FILS IN
  • February 2020: What if the shortest path is not a straight line ? at Journée des doctorant.e.s, University of Montpellier. Abstract: FILS IN