Global existence and future asymptotic behaiviour for solutions of the Einstein-Vlasov - scalar field system with surface symmetry

We prove in the cases of plane and hyperbolic symmetries a global in time existence result in the future for cosmological solutions of the Einstein-Vlasov-scalar field system, with the sources generated by a distribution function and a scalar field, subject to the Vlasov and wave equations respectively. The spacetime is future geodesically complete in the special case of plane symmetry with only a scalar field. Causal geodesics are also shown to be future complete for general homogeneous solutions of the Einstein-Vlasov-scalar field system with plane and hyperbolic symmetry.