It is well-known (and easily shown) that a local quasi-geodesic (for some value of "local") in a $\delta$-hyperbolic space is global (one can compute the constants, as well, from local data). This is false in CAT(0) (think of Euclidean space), but there must be some condition under which it is true. A quick virtual leafing through Bridson-Haefliger brought no joy. (the hidden agenda is figuring out when ping-pong type free subgroups are quasi-isometrically embedded, and with what distortion). Any pointers?
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asked Jun 2, 2015 at 16:48
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$\begingroup$ I know examples of ping-pong type subgruops which are quasi-isometrically embedded, e.g. in mapping class groups, but the methods of proof that I know involve not-properly-discontinuous-but-otherwise-nice actions of the ambient group (or nondistorted subgroups thereof) on $\delta$-hyperbolic spaces. $\endgroup$– Lee MosherCommented Jun 2, 2015 at 19:24
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$\begingroup$ @LeeMosher Yes, as in Kent-Leininger... $\endgroup$– Igor RivinCommented Jun 2, 2015 at 19:53
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$\begingroup$ So the question is about local quasigeodesics ? The title might suggest otherwise. $\endgroup$– HenrikRüpingCommented Jun 3, 2015 at 5:19
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$\begingroup$ @HenrikRüping true, will fix. $\endgroup$– Igor RivinCommented Jun 3, 2015 at 15:09
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1$\begingroup$ @HJRW Well, take a curve made of straight line segments of length $L$ meeting at angle $\pi -\epsilon.$ This will eventually close up into an $2N$-gon (for $\epsilon = \pi/N$), so is not a global quasi-geodesic no matter how large $L$ is, while being a local quasi-geodesic. $\endgroup$– Igor RivinCommented Jun 3, 2015 at 23:39
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Local-to-global results for "Morse quasigeodesics" in symmetric spaces are proven in section 7 of this paper that I wrote with Bernhard Leeb and Joan Porti.
