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 found an article that contains Markov chains on groups: https://arxiv.org/pdf/2111.09837.pdf
- There exists a theorem that refers to Markov properties for groups: https://en.wikipedia.org/wiki/Adian%E2%80%93Rabin_theorem#Markov_property But I think that the Markov properties mentioned on this Wikipedia page don't have to do anything with my problem.

needto understand it in order to understand the paper. $\endgroup$3more comments