# Markov property for groups?

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:

• I think this is a reference to §8.4.A of Gromov's seminal 1987 article "Hyperbolic groups".
– HJRW
Sep 11, 2023 at 15:16
• 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.
– HJRW
Sep 12, 2023 at 13:39
• 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.
– HJRW
Sep 13, 2023 at 8:23
• IMO, this is a very difficult article for a masters' thesis. But of course you should discuss it properly with your mentor.
– HJRW
Sep 14, 2023 at 8:46
• @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. Sep 20, 2023 at 15:25