Qualitative counting closed geodesics
Tóm tắt
We investigate the geometry of word metrics on fundamental groups of manifolds associated with the generating sets consisting of elements represented by closed geodesics. We ask whether the diameter of such a metric is finite or infinite. The first answer we interpret as an abundance of closed geodesics, while the second one as their scarcity. We discuss examples for both cases.
Tài liệu tham khảo
Ballmann, W., Thorbergsson, G., Ziller, W.: Closed geodesics and the fundamental group. Duke Math. J. 48(3), 585–588 (1981)
Bardakov, V., Shpilrain, V., Tolstykh, V.: On the palindromic and primitive widths of a free group. J. Algebra 285(2), 574–585 (2005)
Brandenbursky, M., Kędra, J., Shelukhin, E.: On the autonomous norm on the group of Hamiltonian diffeomorphisms of the torus. Commun. Contemp. Math. 20(2), 1750042 (2018)
Bridson, Martin R., Haefliger, André: Metric spaces of non-positive curvature. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319. Springer-Verlag, Berlin (1999)
Calegari, D.: scl. MSJ Memoirs, vol. 20. Mathematical Society of Japan, Tokyo (2009)
Chinburg, T., Reid, A.W.: Closed hyperbolic \(3\)-manifolds whose closed geodesics all are simple. J. Differ. Geom. 38(3), 545–558 (1993)
Gal, Ś.R., Kędra, J.: On bi-invariant word metrics. J. Topol. Anal. 3(2), 161–175 (2011)
Gal, Światosław R., Kędra, Jarek: Finite index subgroups in chevalley groups are bounded: an addendum to “on bi-invariant word metrics”. arXiv:1808.06376 (2018)
Jones, K.N., Reid, A.W.: Geodesic intersections in arithmetic hyperbolic \(3\)-manifolds. Duke Math. J. 89(1), 75–86 (1997)
Klingenberg, Wilhelm P. A.: Riemannian Geometry, volume 1 of De Gruyter Studies in Mathematics. Walter de Gruyter & Co., Berlin, second edition (1995)
Margulis, G.A.: Discrete subgroups of semisimple Lie groups. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 17. Springer-Verlag, Berlin (1991)
Milnor, J.: A note on curvature and fundamental group. J. Differ. Geometry 2, 1–7 (1968)
Schwarz, A. S.: A volume invariant of coverings. In: Dokl. Akad. Nauk SSSR (N.S.), Vol 105, pp. 32–34 (1955)