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Closed geodesics on non-simply-connected manifolds. (English. Russian original) Zbl 0822.53030

Sib. Math. J. 34, No. 6, 1154-1160 (1993); translation from Sib. Mat. Zh. 34, No. 6, 170-178 (1993).
It is known that the number \(N(t)\) of geometrically distinct geodesics of length \(<t\) on a given Riemannian manifold \(M^ n\), \(n > 1\) satisfies (*) \(N(t) \geq C(t /\ln (t))\) if \(M\) is closed and \(\pi_ 1(M^ n)\) is infinite nilpotent (where the constant \(C\) depends on the metric) [ see V. Bangert and N. Hingston, J. Differ. Geom. 19, 277–282 (1984; Zbl 0545.53036) and W. Ballman, Topology 25, 55–69 (1986; Zbl 0601.53040)]. The author generalizes this result to manifolds with almost solvable fundamental groups, i.e., he proves that the estimate (*) is true if the fundamental group \(\pi_ 1(M)\) contains some normal abelian subgroup of finite index \(G\) such that \(\pi_ 1(M)/G\) is aperiodic (Theorem 1); or there is a normal series \(1 \subset H \subset F \subset G\) such that (1) \(G\) is a subgroup of finite index in \(\pi_ 1(M)\), (2) \(H\) is a nonzero commutative subgroup, (3) \(G/F\) is an infinite cyclic group (Theorem 2); or if \(\pi_ 1(M)\) contains a solvable subgroup of finite index (Theorem 3).

MSC:

53C22 Geodesics in global differential geometry
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[1] I. A. Taîmanov, ?Closed extremals on two-dimensional manifolds,? Uspekhi Mat. Nauk,47, No. 2, 143-185 (1992).
[2] V. Bangert and N. Hingston, ?Closed geodesics on manifolds with infinite abelian fundamental group,? J. Differential Geom.,19, No. 2, 277-282 (1984). · Zbl 0545.53036
[3] M. Tanaka, Closed Geodesics on Compact Riemannian Manifolds with Infinite Fundamental Group. I [Preprint], Tokai Univ. (1983).
[4] I. A. Taîmanov, ?Closed geodesics on non-simply-connected manifolds,? Uspekhi Mat. Nauk,40, No. 6, 157-158 (1985).
[5] W. Ballmann, ?Geschlossene Geodätische auf Mannigfaltigkeiten mit unendlicher Fundamentalgruppe,? Topology,25, No. 1, 55-69 (1986). · Zbl 0601.53040 · doi:10.1016/0040-9383(86)90005-4
[6] I. K. Babenko, ?Closed geodesics, asymptotic volume and growth characteristics of groups,? Izv. Akad. Nauk SSSR Ser. Mat.,52, No. 4, 675-711 (1988). · Zbl 0657.53024
[7] M. Gromov, ?Groups of polynomial growth and expanding maps,? Inst. Hautes Études Sci. Publ. Math., No. 53, 53-73 (1981). · Zbl 0474.20018 · doi:10.1007/BF02698687
[8] V. Bangert and W. Klingenberg, ?Homology generated by iterated closed geodesics,? Topology,22, No. 4, 379-388 (1983). · Zbl 0525.58015 · doi:10.1016/0040-9383(83)90033-2
[9] Sze-tsen Hu, Homotopy Theory [Russian translation], Mir, Moscow (1964).
[10] A. G. Kurosh, Group Theory [in Russian], Nauka, Moscow (1967).
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