Filtered Hopf algebras and counting geodesic chords

We prove lower bounds on the growth of certain filtered Hopf algebras by means of a Poincaré–Birkhoff–Witt type theorem for ordered products of primitive elements. When applied to the loop space homology algebra endowed with a natural length-filtration, these bounds lead to lower bounds for the number of geodesic paths between two points. Specifically, given a closed manifold  \(M\) whose universal covering space is not homotopy equivalent to a finite complex and whose fundamental group has polynomial growth, for any Riemannian metric on  \(M\) , any pair of non-conjugate points \(p,q \in M\) , and every component  \({\mathcal C}\) of the space of paths from  \(p\) to  \(q\) , the number of geodesics in  \({\mathcal C}\) of length at most  \(T\) grows at least like \(e^{\sqrt{T}}\) . Using Floer homology, we extend this lower bound to Reeb chords on the spherisation of  \(M\) , and give a lower bound for the volume growth of the Reeb flow.