Constant mean curvature foliations of simplicial flat spacetimes

Benedetti and Guadagnini [5] have conjectured that the constant mean curvature foliation M-tau in a 2 + 1 dimensional flat spacetime V with compact hyperbolic Cauchy surfaces satisfies lim(tau-->-infinity) l(M tau) = s(T), where lM(tau) and s(T) denote the marked length spectrum of M-tau and the marked measure spectrum of the R-tree T, dual to the measured foliation corresponding to the translational part of the holonomy of V, respectively. We prove that this is the case for n + 1 dimensional, n >= 2, simplicial flat spacetimes with compact hyperbolic Cauchy surface. A simplicial spacetime is obtained from the Lorentz cone over a hyperbolic manifold by deformations corresponding to a simple measured foliation.