Geschlossene Geodätische auf Mannigfaltigkeiten mit unendlicher Fundamentalgruppe. (Closed geodesics on manifolds with infinite fundamental group) (Q1081131)

Let N(t) denote the number of the geometrically distinct closed geodesics of length less than t on a compact Riemann manifold. \textit{M. Gromov} [Three remarks on geodesic dynamics, Preprint S. N. Brook] obtained an evaluation of the asymptotic behavior of N(t) in the case that the manifold is homeomorphic to \(S^ 1\times V_ 0\), where \(V_ 0\) is a simply connected manifold, as follows \[ \liminf_{t\rightharpoonup \infty}N(t)/\ln (t)>0, \] and if \(V_ 0\) is a Lie group in addition, then \[ \limsup_{t\rightharpoonup \infty}N(t)/t^ 2>0. \] The author generalizes the results above the Theorem 1. If V is homeomorphic to \(S^ 1\times V_ 0\), where \(V_ 0\) is a simply connected manifold, then \[ \liminf_{t\to \infty}N(t)\ln (t)/t^ 2>0\quad and\quad \limsup_{t\to \infty}N(t)/t^ 2>0. \] For a subgroup Z of the fundamental group of V, a function \(N^ Z(t)\) is defined in analogy to N(t) to be the number of closed geodesics of length less than t, which represent a non zero element in Z. The author generalizes Gromov's evaluation also for this function. Theorem 2. If the fundamental group is almost nilpotent and does not coincide with the infinite cyclic group, then \[ \liminf_{t\to \infty}N^ Z(t)\ln (t)/t>0 \] provided that Z is infinite cyclic. As a matter of fact, the assumption that the fundamental group is not infinite cyclic is superfluous, because it is used to avoid the case where Z agrees with the fundamental group and in this case the evaluation is obtained by \textit{V. Bangert} and \textit{N. Hingston} [J. Differ. Geom. 19, 277-282 (1984; Zbl 0545.53036)].