Aebischer, Beat; Hong, Sungbok; McCullough, Darryl Recurrent geodesics and controlled concentration points. (English) Zbl 0810.30034 Duke Math. J. 75, No. 3, 759-774 (1994). A complete hyperbolic manifold is a quotient \(M= B/\Gamma\), where \(B\subset\mathbb{R}^ m\) is the open unit ball with the Riemannian metric \(ds= {2| dx|\over 1-| x|^ 2}\), i.e. the Poincaré model of hyperbolic space, and \(\Gamma\) is a discrete torsion free group of hyperbolic isometries. An oriented geodesic \(\beta: \mathbb{R}\to M\) parametrized by arclength is recurrent if for all (equivalently for some) \(t_ 0\in \mathbb{R}\) there is a sequence \(t_ n\to \infty\) such that \(\beta'(t_ n)\) converges to \(\beta'(t_ 0)\) in the unit tangent bundle \(T_ 1 M\). It is shown that a geodesic \(\beta\) is recurrent if and only if it is approximable by closed geodesics, i.e. for all \(t_ 0\in \mathbb{R}\) and \(\varepsilon> 0\) there exists a closed geodesic \(\gamma: \mathbb{R}\to M\) such that \(\rho(\gamma(t),\beta(t_ 0+ t))\leq \varepsilon\) for \(0\leq t\leq\text{length}(\gamma)\). (Obviously, \(\text{length}(\gamma)\to \infty\) when \(\varepsilon\to 0\), unless \(\beta\) is closed.) As a corollary, the recurrent geodesics form a \(G_ \delta\)-subset \(\mathcal R\) of the space of all geodesics (which can be identified with \(T_ 1 M\)), hence \(\mathcal R\) is topologically complete.The endpoints of lifts to \(B\) of recurrent geodesics in \(M\) are characterized as the special limit points of \(\Gamma\) mentioned in the title. A limit point of \(\Gamma\) is a controlled concentration point \((p\in \Lambda_{ccp})\) if there is a neighborhood \(U\subset \partial B\) of \(p\) such that for every neighborhood \(V\) of \(p\) there exists \(g\in \Gamma\) satisfying \(g(U)\subseteq V\) and \(p\in g(V)\). The paper gives several other characterizations of controlled concentration points similar to characterizations of conical limit points. For instance, \(p\in \Lambda_{ccp}\) if and only if \(\exists(g_ n)\subset\Gamma: g_ n(p)\to p\) and \(g_ n(0)\to r\in \partial B\backslash\{p\}\). Clearly, \(\Lambda_{ccp}\) is a subset of the set of conical limit points \(\Lambda_ c\), and it follows from known results that \(\Lambda_{ccp}\) has full or zero measure according to whether \(\Gamma\) is of divergence or of convergence type. A comparison of the above definition of \(\Lambda_{ccp}\) with the following ‘concentration characterization’ of \(\Lambda_ c\) (which the third author found after the paper was published) nicely shows the difference between these two kinds of limit points: \(p\in \Lambda_ c\) if and only if there exist open neighborhoods \(U_ 1\) and \(U_ 2\) of \(p\) such that \(\overline U_ 2\subset U_ 1\) and for every neighborhood \(V\) of \(p\) there exists \(g\in \Gamma\) with \(g(U_ 1)\subseteq V\) and \(p\in g(U_ 2)\).The last section contains a Schottky group example showing that \(\Lambda_ c\backslash\Lambda_{ccp}\) can be uncountable. (To correct a misprint in Proposition 1.8, replace ‘distance \(\beta\)’ by ‘distance \(\delta\)’ just before (i)). Reviewer: B.Aebischer Cited in 1 ReviewCited in 1 Document MSC: 30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization) Keywords:Möbius group; limit point; discrete group; recurrent geodesics; Schottky group PDFBibTeX XMLCite \textit{B. Aebischer} et al., Duke Math. J. 75, No. 3, 759--774 (1994; Zbl 0810.30034) Full Text: DOI References: [1] B. Aebischer, The limiting behavior of sequences of Möbius transformations , Math. Z. 205 (1990), no. 1, 49-59. · Zbl 0686.40002 [2] B. Aebischer, The limiting behavior of sequences of quasiconformal mappings , Canad. Math. Bull. 33 (1990), no. 4, 494-502. · Zbl 0699.30016 [3] S. Agard, A geometric proof of Mostow’s rigidity theorem for groups of divergence type , Acta Math. 151 (1983), no. 3-4, 231-252. · Zbl 0532.30038 [4] L. V. Ahlfors, Möbius transformations in several dimensions , Ordway Professorship Lectures in Math., University of Minnesota School of Mathematics, Minneapolis, Minn., 1981. · Zbl 0517.30001 [5] A. Beardon, The geometry of discrete groups , Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1983. · Zbl 0528.30001 [6] A. Beardon and B. Maskit, Limit points of Kleinian groups and finite sided fundamental polyhedra , Acta Math. 132 (1974), 1-12. · Zbl 0277.30017 [7] A. Casson and S. Bleiler, Automorphisms of surfaces after Nielsen and Thurston , London Mathematical Society Student Texts, vol. 9, Cambridge University Press, Cambridge, 1988. · Zbl 0649.57008 [8] J. Dugundji, Topology , Allyn and Bacon Inc., Boston, Mass., 1966. · Zbl 0144.21501 [9] J. B. Garnett, F. W. Gehring, and P. W. Jones, Quasiconformal groups and the conical limit set , Holomorphic functions and moduli, Vol. II (Berkeley, CA, 1986) ed. E. Drasin, et al., Math. Sci. Res. Inst. Publ., vol. 11, Springer, New York, 1988, pp. 59-67. · Zbl 0654.30013 [10] F. W. Gehring and G. J. Martin, Discrete quasiconformal groups. I , Proc. London Math. Soc. (3) 55 (1987), no. 2, 331-358. · Zbl 0628.30027 [11] S. Hong, Controlled concentration points and groups of divergence type , to appear in proceedings of Low-dimensional Topology Conference, Knoxville, Tennessee, 1992. · Zbl 0872.22007 [12] B. Maskit, Kleinian Groups , Grundlehren Math. Wiss., vol. 287, Springer-Verlag, Berlin, 1987. [13] D. McCullough, Weak concentration points for discrete groups of Möbius transformations , to appear in Illinois J. Math. · Zbl 0805.20041 [14] D. McCullough, Concentration points for Fuchsian groups , · Zbl 0959.20046 [15] P. Nicholls, Transitive horocycles for Fuchsian groups , Duke Math. J. 42 (1975), 307-312. · Zbl 0348.30018 [16] P. Nicholls, The Ergodic Theory of Discrete Groups , London Mathematical Society Lecture Note Series, vol. 143, Cambridge University Press, Cambridge, 1989. · Zbl 0674.58001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.