Constant mean curvature foliations of simplicial flat spacetimes

Benedetti and Guadagnini have conjectured that the marked lenght spectrum of the constant mean curvature foliation $M_\tau$ in a 2+1 dimensional flat spacetime $V$ with compact hyperbolic Cauchy surfaces converges, in the direction of the singularity, to that of the marked measure spectrum of the R-tree dual to the measured foliation corresponding to the translational part of the holonomy of $V$. We prove that this is the case for $n+1$ dimensional, $n \geq 2$, {\em 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.