Geometric transition from hyperbolic to Anti-de Sitter structures in dimension four

Stefano Riolo (1) , Andrea Seppi (2)

1 UNINE - Université de Neuchâtel = University of Neuchatel
2 IF - Institut Fourier

We provide the first examples of geometric transition from hyperbolic to Anti-de Sitter structures in dimension four, in a fashion similar to Danciger's three-dimensional examples. The main ingredient is a deformation of hyperbolic 4-polytopes, discovered by Kerckhoff and Storm, eventually collapsing to a 3-dimensional ideal cuboc-tahedron. We show the existence of a similar family of collapsing Anti-de Sitter polytopes, and join the two deformations by means of an opportune half-pipe orbifold structure. The desired examples of geometric transition are then obtained by gluing copies of the polytope.

Domaines: Topologie géométrique [math.GT]

Stefano Riolo, Andrea Seppi. Geometric transition from hyperbolic to Anti-de Sitter structures in dimension four. Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, 2022, XXIII (1), pp.115-176. ⟨10.2422/2036-2145.202005_031⟩. ⟨hal-03000924⟩