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

Abstract

We study the link between Anosov representations and (G, X)-structure on spacetimes with $G = O_0(2,n)$ and X is the Einstein universe $Ein_{1,n-1}$ ($n \ge 3$), namely conformally flat spacetimes. Let $\varGamma$ be a Gromov hyperbolic group and let $\rho : \varGamma \rightarrow O_0(2,n)$ be a $P_1$-Anosov representation as defined in [14]. We suppose that the limit set is an acausal subset in $Ein_{1,n-1}$. Our first result is that $\rho$ is the holonomy of a globally hyperbolic spatially compact maximal (abbrev. GHCM) conformally flat spacetime of dimension n. This comes in the continuity of Barbot-Mérigot work in anti-de Sitter geometry Barbot and Mérigot (Groups Geometry Dynamics 6(3): 441–483 2012): they proved that when the limit set is a topological $(n-1)$-sphere then $\rho$ is the holonomy of a GHCM spacetime locally modeled on the anti-de Sitter spacetime $AdS_{1,n}$. Our second result extends this last one and state that, except in this particular case, $\rho$ is the holonomy of a strongly causal AdS-spacetime of dimension $(n+1)$ which is a BTZ-black hole Banados et al. (Phys Rev D Part Fields 48(4):1506–1525 1993), Banados et al. (Phys Rev Lett 69(13):1849–1851 1992), Barbot T (Adv Theor Math Phys 12:1209–1257 2005).