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?

  • $\begingroup$ 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 Mosher Jun 2 '15 at 19:24
  • $\begingroup$ @LeeMosher Yes, as in Kent-Leininger... $\endgroup$ – Igor Rivin Jun 2 '15 at 19:53
  • $\begingroup$ So the question is about local quasigeodesics ? The title might suggest otherwise. $\endgroup$ – HenrikRüping Jun 3 '15 at 5:19
  • $\begingroup$ @HenrikRüping true, will fix. $\endgroup$ – Igor Rivin Jun 3 '15 at 15:09
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    $\begingroup$ @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 Rivin Jun 3 '15 at 23:39

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.

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