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)

Urs Frauenfelder; Felix Schlenk

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arXiv · 1309.6481

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)

Urs Frauenfelder·Felix Schlenk

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Abstract

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.

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BibTeX RIS

Urs Frauenfelder, Felix Schlenk. 2013-09-25. 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). https://arxiv.org/abs/1309.6481

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