  We give a uniform lower bound for the polynomial complexity of all Reeb flows
on the spherization (S*M,\xi) over a closed manifold. Our measure for the
dynamical complexity of Reeb flows is slow volume growth, a polynomial version
of topological entropy, and our uniform bound is in terms of the polynomial
growth of the homology of the based loops space of M. As an application, we
extend the Bott--Samelson theorem from geodesic flows to Reeb flows: If
(S*M,\xi) admits a periodic Reeb flow, or, more generally, if there exists a
positive Legendrian loop of a fibre S*_q M, then M is a circle or the
fundamental group of M is finite and the integral cohomology ring of the
universal cover of M is the one of a compact rank one symmetric space.
