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Let $M$ be a maximal globally hyperbolic Cauchy compact flat spacetime ofdimension 2+1, admitting a Cauchy hypersurface diffeomorphic to a compacthyperbolic manifold. We study the asymptotic behaviour of level sets ofquasi-concave time functions on $M$. We give a positive answer to a conjectureof Benedetti and Guadagnini in \cite{MR1857817}. More precisely, we prove thatthe level sets of such a time function converge in the Hausdorff-Gromovequivariant topology to a real tree. Moreover, this limit does not depend onthe choice of the time function.

Sur la g\'eom\'etrie de la singularit\'e initiale des espaces-temps plats globalement hyperboliques