  We give lower bounds for the growth of the number of Reeb chords and for the
volume growth of Reeb flows on spherizations over closed manifolds M that are
not of finite type, have virtually polycyclic fundamental group, and satisfy a
mild assumption on the homology of the based loop space.
  For the special case of geodesic flows, these lower bounds are:
  (i) For any Riemannian metric on M, any pair of non-conjugate points p,q in
M, and every component C of the space of paths from p to q, the number of
geodesics in C of length at most T grows at least like e^{\sqrt T}.
  (ii) The exponent of the volume growth of any geodesic flow on M is at least
1/2.
  We obtain these results by combining new algebraic results on the growth of
certain filtered Hopf algebras with known results on Floer homology.
