Geodesics and Visual boundary of Horospherical Products

Tom Ferragut (1)

1 UM - Université de Montpellier

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 H p and H q , we study the geometry of their horospherical product H ∶= H p ⋈ H q through a description of its geodesics. Specically we introduce a large family of distances on H p ⋈ H q. 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.

Mots clés: Horocyclic product; Horospherical product; Gromov Hyperbolicity; Busemann space; Geodesic; Visual boundary

Domaines: Géométrie métrique [math.MG]

Tom Ferragut. Geodesics and Visual boundary of Horospherical Products. 2020. ⟨hal-02933134v1⟩