Filtered Hopf algebras and counting geodesic chords

Abstract

We prove lower bounds on the growth of certain filtered Hopf algebras by means of a Poincare–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.