Anosov representations as holonomies of globally hyperbolic spatially compact conformally flat spacetimes

TL;DR

This paper demonstrates that certain Anosov representations into O0(2,n) correspond to holonomies of specific globally hyperbolic, conformally flat spacetimes, linking geometric group actions with Lorentzian geometry.

Contribution

It establishes a correspondence between Anosov representations with negative limit sets and the holonomy groups of spatially compact, globally hyperbolic conformally flat spacetimes.

Findings

Anosov representations act properly discontinuously on causal geodesics.

The associated spacetime is the union of two conformal copies of a strongly causal AdS spacetime.

The spacetime can contain black hole regions when the limit set is not a topological sphere.

Abstract

Anosov representations were introduced by F. Labourie [18] for fundamental groups of closed negatively curved surfaces, and generalized by O. Guichard and A. Wienhard [19] to representations of arbitrary Gromov hyperbolic groups into real semisimple Lie groups. In this paper, we focus on Anosov representations into the identity component O0(2, n) of O(2, n) for n $\ge$ 2. Our main result is that any Anosov representation with negative limit set as defined in [8] is the holonomy group of a spatially compact, globally hyperbolic maximal (abbrev. CGHM) conformally flat spacetime. The proof of the spatial compactness needs a particular care. The key idea is to notice that for any spacetime M , the space of lightlike geodesics of M is homeomorphic to the unit tangent bundle of a Cauchy hypersurface of M. For this purpose, we introduce the space of causal geodesics containing timelike and lightlike geodesics of anti-de Sitter space and lightlike geodesics of its conformal boundary: the Einstein spacetime. The spatial compactness is a consequence of the following theorem : Any Anosov representation acts properly discontinuously by isometries on the set of causal geodesics avoiding the limit set; besides, this action is cocompact. It is stated in a general setting by O. Guichard, F. Kassel and A. Wienhard in [11]. We can see this last result as a Lorentzian analogue of the action of convex cocompact Kleinian group on the complementary of the limit set in H n. Lastly, we show that the conformally flat spacetime in our main result is the union of two conformal copies of a strongly causal AdS-spacetime with boundary which contains-when the limit set is not a topological (n -- 1)-sphere-a globally hyperbolic region having the properties of a black hole as defined in [2], [3], [4].