HomeData scienceRevisiting Anosov representations part5(Machine Studying 2024) | by Monodeep Mukherjee | Feb,...

Revisiting Anosov representations part5(Machine Studying 2024) | by Monodeep Mukherjee | Feb, 2024

  1. Maximal and Borel Anosov representations in Sp(4,R)(arXiv)

Creator : Colin Davalo

Summary : We show that any Borel Anosov representations of a floor group into Sp(4,R) that has maximal Toledo invariant have to be Hitchin. We additionally show {that a} illustration of a floor group into Sp(2n,R) that’s {n−1,n}-Anosov is maximal if and provided that it satisfies the hyperconvexity property Hn.

2.Anosov representations as holonomies of worldwide hyperbolic spatially compact conformally flat spacetimes (arXiv)

Creator : Rym Smai

Summary : Anosov representations have been launched by F. Labourie [18] for elementary teams of closed negatively curved surfaces, and generalized by O. Guichard and A. Wienhard [19] to representations of arbitrary Gromov hyperbolic teams into actual semisimple Lie teams. On this paper, we deal with Anosov representations into the identification element O0(2, n) of O(2, n) for n ≥ 2. Our principal result’s that any Anosov illustration with unfavourable restrict set as outlined in [8] is the holonomy group of a spatially compact, globally hyperbolic maximal (abbrev. CGHM) conformally flat spacetime. The proof of the spatial compactness wants a specific care. The important thing thought is to note that for any spacetime M , the area of lightlike geodesics of M is homeomorphic to the unit tangent bundle of a Cauchy hypersurface of M. For this function, we introduce the area of causal geodesics containing timelike and lightlike geodesics of anti-de Sitter area and lightlike geodesics of its conformal boundary: the Einstein spacetime. The spatial compactness is a consequence of the next theorem : Any Anosov illustration acts correctly discontinuously by isometries on the set of causal geodesics avoiding the restrict set; in addition to, this motion is cocompact. It’s acknowledged in a basic setting by O. Guichard, F. Kassel and A. Wienhard in [11]. We will see this final outcome as a Lorentzian analogue of the motion of convex cocompact Kleinian group on the complementary of the restrict set in H n. Lastly, we present that the conformally flat spacetime in our principal result’s the union of two conformal copies of a strongly causal AdS-spacetime with boundary which contains-when the restrict set shouldn’t be a topological (n — 1)-sphere-a globally hyperbolic area having the properties of a black gap as outlined in [2], [3], [4]

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