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Topological mixing in $CAT\left(-1\right)$-spaces

Authors: Charalambos Charitos and Georgios Tsapogas
Journal: Trans. Amer. Math. Soc. 354 (2002), 235-264
MSC (2000): Primary 57M20; Secondary 53C23
Published electronically: August 21, 2001
MathSciNet review: 1859274
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Abstract: If $X$ is a proper $CAT\left( -1\right) $-space and $\Gamma$ a non-elementary discrete group of isometries acting properly discontinuously on $X,$ it is shown that the geodesic flow on the quotient space $Y=X/\Gamma$ is topologically mixing, provided that the generalized Busemann function has zeros on the boundary $\partial X$ and the non-wandering set of the flow equals the whole quotient space of geodesics $GY:=GX/\,\Gamma$ (the latter being redundant when $Y$ is compact). Applications include the proof of topological mixing for (A) compact negatively curved polyhedra, (B) compact quotients of proper geodesically complete $CAT\left( -1\right) $-spaces by a one-ended group of isometries and (C) finite $n$-dimensional ideal polyhedra.

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  • 1. D.V. Anosov, Geodesic flows on closed Riemannian manifolds with negative curvature, Proc. Steklov Instit. Math. 90, Amer. Math. Soc., Providence, Rhode Island, 1969. MR 39:3527
  • 2. W. Ballmann, Lectures on Spaces of Non-positive Curvature, Birkhäuser, 1995. MR 97a:53053
  • 3. W. Ballman, E. Ghys, A. Haefliger, P. de la Harpe, E. Salem, R. Strebel and M. Troyanov, Sur les groups hyperboliques d'aprés Gromov (Seminaire de Berne), édité par E. Ghys et P. de la Harpe, (a paraitre chez Birkhäuser), 1990.
  • 4. M. Bourdon, Structure conforme au bord et flot géodésique d'un CAT $\left( -1\right) $-espace, Enseign. Math. 41, (1995), 63-102. MR 96f:58120
  • 5. B. H. Bowditch, Boundaries of geometrically finite groups, Math. Z. 230 (1999) no. 3, 509-527. MR 2000b:20049
  • 6. B. H. Bowditch, Connectedness properties of limit sets, Trans. Amer. Math. Soc. 351 (1999) 3673-3686. MR 2000d:20056
  • 7. M. Bridson, Geodesics and curvature in metric simplicial complexes in Group Theory from a Geometrical Viewpoint, (ICTP, Trieste, Italy, March 26-April 6, 1990), E.Ghys and A.Haefliger, eds., (1991). MR 94c:57040
  • 8. Ch. Champetier, Petite simplification dans les groupes hyperboliques, Ann. Fac. Sci. Toulouse, VI. Ser., Math. 3, No.2, 161-221, (1994). MR 95e:20050
  • 9. C. Charitos, A. Papadopoulos, The geometry of ideal polyhedra, to appear in Glasgow Journal of Mathematics.
  • 10. C. Charitos, G. Tsapogas, Geodesic flow on ideal polyhedra, Canad. J. Math. 49 (4) 1997, pp. 696-707. MR 98e:58133
  • 11. C. Charitos, Closed geodesics in ideal polyhedra of dimension 2, Rocky Mountain Journal of Mathematics, Vol. 26, no. 2 (1996), pp. 507-521. MR 97d:57014
  • 12. C. Charitos, G. Tsapogas, Complexity of Geodesics on 2-dimensional ideal polyhedra and isotopies, Math. Proc. Camb. Phil. Soc., Vol. 121 (1997), pp. 343-358. MR 98b:57006
  • 13. C. Charitos, G. Tsapogas, Approximation of recurrence in negatively curved metric spaces, Pacific J. Math. 195 (2000), 67-79. CMP 2001:01
  • 14. M. Coornaert, Sur les groupes proprement discontinus d'isométries des espaces hyperboliques au sens de Gromov, Thèse U.L.P., Publication de l'IRMA. MR 92i:57032
  • 15. M. Coornaert, T. Delzant, A. Papadopoulos, Gé ométrie et théorie des groupes, Lecture Notes in Mathematics, vol.1441, Springer-Verlag, (1990). MR 92f:57003
  • 16. P. Eberlein, Geodesic flows on negatively curved manifolds II, Trans. Amer. Math. Soc. 178 (1973), pp. 57-82. MR 47:2636
  • 17. M. Gromov, Hyperbolic groups, in Essays in Group Theory, MSRI Publ. 8, Springer, 1987, pp. 75-263. MR 89e:20070
  • 18. S. Hersonsky, F. Paulin, On the rigidity of discrete isometry groups of negatively curved spaces, Comm. Math. Helv. 72 (1997), pp. 349-388. MR 98h:58105
  • 19. V.A. Kaimanovich, Ergodicity of harmonic invariant measures for the geodesic flow on hyperbolic spaces, J. Reine Angew. Math. 445 (1994), pp. 57-103. MR 95g:58130
  • 20. G. Moussong, Hyperbolic Coxeter groups, Doctoral Dissertation, Ohio State University, 1988.
  • 21. F. Paulin, Constructions of hyperbolic groups via hyperbolization of polyhedra, in Group Theory from a Geometrical Viewpoint, (ICTP, Trieste, Italy, March 26-April 6, 1990), E. Ghys and A. Haefliger, eds., (1991). MR 93d:57005
  • 22. J. Ratcliffe, Foundations of hyperbolic geometry, GTM, Springer-Verlag, 1994. MR 95j:57011
  • 23. W.P. Thurston, The Geometry and Topology of Three-manifolds, Lecture notes, Princeton University, Princeton, NJ (1979). 2

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Additional Information

Charalambos Charitos
Affiliation: Department of Mathematics, Agricultural University of Athens, 75 Iera Odos, Athens, Greece

Georgios Tsapogas
Affiliation: Department of Mathematics, University of The Aegean, Karlovassi, Samos 83200, Greece

Keywords: $CAT\left( -1\right)$-space, mixing, geodesic flow, negatively curved polyhedra
Received by editor(s): August 13, 1999
Received by editor(s) in revised form: May 18, 2000
Published electronically: August 21, 2001
Additional Notes: This research was supported in part by Research Unit Grant 470
Article copyright: © Copyright 2001 American Mathematical Society

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