Exponential Growth of Spaces without Conjugate Points

An n-dimensional polyhedral space is a length space M (with intrinsic metric) triangulated into n-simplexes with smooth Riemannian metrics. In the definitions below, we assume that the triangulation is fixed. The boundary of M is the union of the (n− 1)-simplexes of the triangulation that are adjacent to only one (n− 1)-simplex. As usual, a geodesic in M is a naturally parametrized locally shortest curve defined on an interval. We say that M has no conjugate points if any two points in the universal covering space M̃ of M are joined by a unique geodesic. We say that the volume entropy of M̃ is positive if the volume of metric balls in M̃ has at least exponential growth. Now, we state the main result of this paper.