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Transactions of the American Mathematical Society

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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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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