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My question again refers to the following article:

  • Koji Fujiwara, Zlil Sela, The rates of growth in a hyperbolic group, Invent. math. 233 (2023) pp 1427–1470, doi:10.1007/s00222-023-01200-w, arXiv:2002.10278.

On page 34 we have the following:

To prove the proposition, we follow the proof of proposition 3.2, although unlike finite generating sets of the ambient hyperbolic group $\Gamma$, Cayley graphs of subgroups of $\Gamma$ with respect to their finite generating sets are not guaranteed to have the Markov property.

My guess is that the "Markov property" refers to the following text further down on page 34/35:

In general, with the generating set $S_{n_0}$ and the subgroup $H_{n_0}$ we can not associate a finite state automata, that constructs a single geodesic from the identity to each element in the Cayley graph of $H_{n_0}$, as we did in the proof of proposition 3.2.

So the Markov property could be that in the group case (in contrast to the subgroup case) one can associate a finite state automata [...] as it is done on page 18/19 in the article. But I am not really sure. Of course I know what Markov chains are in probability theory but this didn't help me so much. My tutor couldn't give me an answer to my question. I also wrote an email to the authors but didn't get an answer.

Maybe two remarks that could help:

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    $\begingroup$ I think this is a reference to §8.4.A of Gromov's seminal 1987 article "Hyperbolic groups". $\endgroup$
    – HJRW
    Commented Sep 11, 2023 at 15:16
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    $\begingroup$ It is a bit cryptic. But the quoted text looks like motivation, rather than the details of the proof, so presumably you don't need to understand it in order to understand the paper. $\endgroup$
    – HJRW
    Commented Sep 12, 2023 at 13:39
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    $\begingroup$ Although I don't know your mathematical background, I think it is a LOT to try to read that article from scratch. Sela wrote 1000s of pages about limit groups already, and familiarity with much of that machinery is taken for granted. Plus the theory of hyperbolic groups etc etc. $\endgroup$
    – HJRW
    Commented Sep 13, 2023 at 8:23
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    $\begingroup$ IMO, this is a very difficult article for a masters' thesis. But of course you should discuss it properly with your mentor. $\endgroup$
    – HJRW
    Commented Sep 14, 2023 at 8:46
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    $\begingroup$ @TheMathematician I think the tagging is wrong. This has little to do with probability, but rather with symbolic dynamics. Also, one should emove rings and algebras. $\endgroup$
    – M. Dus
    Commented Sep 20, 2023 at 15:25

1 Answer 1

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Yes the Markov property is what is described in the text further down.

The terminology Markov is not directly related to Markov chains in probability theory. A group is called Markov if there exists a finite state automaton with an initial state and a labeling of the edges by generating elements, such that the application that maps a word in the language generated by the automaton to the corresponding element in the group via the labeling is a bijection.

This terminology is not used anymore and people use "automatic groups" instead. There is a rich body of work on automatic group and the standard reference to start with is the book of Cannon Epstein Holt Paterson and Thurston : Word processing and group theory.

Now the terminology Markov property goes back to the foundations of automatic groups and is indeed inspired by the machinery of Markov chains. This is very well explained in the introduction of Chapter 9 of Ghys and de la Harpe's book : sur les groupes hyperboliques d'après Mickael Gromov (in French).

Note that the terminology Markov is also common in dynamical systems and especially in symbolic dynamics. Subshifts of finite type are sometimes called topological Markov shifts. This is relevant here, as the automaton in the above definition yields a subshift of fintie type encoding elements of the group.

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  • $\begingroup$ Thanks for your answer. Maybe one more question: Do you know an english reference in which the term "Markov property" is defined and used in the same way Fujiwara and Sela use it in their article? I understood you that way that this is not the case in the book of Epstein but I dont have access to it either way. $\endgroup$
    – TheMathematician
    Commented Sep 20, 2023 at 12:24
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    $\begingroup$ @TheMathematician Yes, Calegari's notes called The ergodic theory of hyperbolic groups are very good.Section 3 is especiallly what you need. $\endgroup$
    – M. Dus
    Commented Sep 20, 2023 at 15:18
  • $\begingroup$ I was looking this up and I am not sure. Apparently it is names after Andrey Markov Jr (son of Andrey Markov Senior) who worked on things like these. $\endgroup$
    – Nishant Chandgotia
    Commented Jul 16 at 6:46
  • $\begingroup$ @NishantChandgotia I'm interested in your sources. In my opinion, the machinery of topological Markov shifts is really inspired by the machinery of probabilistic Markov chains $\endgroup$
    – M. Dus
    Commented Jul 16 at 9:12
  • $\begingroup$ Dear @M.Dus for sure the name topological Markov chains comes from Markov chains! You are absolutely correct about that. I was talking about the Markov property for groups. For sure Andrey Markov Jr was influenced by the philosophical outlook of his father but it would seem that the name perhaps comes from his work rather than his father's. In other words, while the property itself may draw inspiration from the philosophy of Markov chains, the name perhaps comes from the creator. Of course I don't have a further reference for this. It is just my feeling! $\endgroup$
    – Nishant Chandgotia
    Commented Jul 17 at 14:57

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