I was wondering if there is a good notion of flows in word-hyperbolic groups (maybe I should say flows in the Cayley graphs of word-hyperbolic groups).

More precisely, I wonder if there is an analogue of either quasi-geodesic flow or pseudo-Anosov flow in hyperbolic 3-manifolds in the setting of word-hyperbolic groups. Any help will be appreciated!

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    $\begingroup$ I don't think it's quite what you're looking for, but Mineyev came up with a definition of the geodesic flow-space of a word-hyperbolic group. See, for instance, Subsection 3.4.2. of Danny Calegari's book scl for a summary of its properties. $\endgroup$ – HJRW Dec 29 '12 at 9:29
  • $\begingroup$ As you said, it seems slightly different from what I am looking for, but still very interesting. I will take a look at the Mineyev's orginal paper too. Thank you for your help! $\endgroup$ – Harry Baik Dec 30 '12 at 3:29

I don't have a general answer, but I have a specific and interesting special case.

Let $\Phi : F_n \to F_n$ be an automorphism of the finite rank free group $F_n$ which is atoroidal meaning that no element of $F_n$ has a periodic conjugacy class. By a theorem of Brinkmann MR1800064 following a theorem of Bestvina and Feighn, the group mapping torus $M_\phi = \langle F_n, t | t g t^{-1} = \Phi(g) \rangle$ is word hyperbolic. If furthermore $f : G \to G$ is a train track map representing $\Phi$ (in the sense of Bestvina and Handel) then the suspension semiflow of $f$ may be regarded as a flow on $M_\phi$, and this flow has many interesting properties that are closely analogous to the suspension flow of a pseudo-Anosov surface homeomorphism. Some properties of this construction were worked out in the 2002 Rutgers thesis of my student Zhifeng Wang entitled "Mapping tori of outer automorphisms of free groups". There may also some published literature on this construction, but I am less sure about that. Wang's thesis is not published other than in the Rutgers library.

  • $\begingroup$ Thank you so much, Lee. This is very interesting. Do we know anything about the space of flow lines or the boundary of it in this case? $\endgroup$ – Harry Baik Dec 30 '12 at 3:23

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