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)

2013/09/25 by Urs Frauenfelder, Frauenfelder, Urs, Felix Schlenk +1
Mathematics · #37C35 #53D25 #57T25 #Algebraic Topology (math.AT) #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Primary 16T05 #Secondary 37B40 #Symplectic Geometry (math.SG) #math.AT #math.DG #math.DS #math.SG #msc:16T05 #msc:37B40 #msc:37C35 #msc:53D25 #msc:57T25

paper · pdf · doi:10.48550/arxiv.1309.6481

21 pages

arxiv created 2013/09/25 · arxiv updated 2013/09/26

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^{√ 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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