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We give lower bounds for the growth of the number of Reeb chords and for thevolume growth of Reeb flows on spherizations over closed manifolds M that arenot of finite type, have virtually polycyclic fundamental group, and satisfy amild 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 inM, and every component C of the space of paths from p to q, the number ofgeodesics 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 least1/2.  We obtain these results by combining new algebraic results on the growth ofcertain filtered Hopf algebras with known results on Floer homology.

Healthy vector spaces and spicy Hopf algebras (with applications to the  growth rate of geodesic chords and to intermediate volume growth on manifolds  of non-finite type)