Cohomological dimension of Markov compacta

We rephrase Gromov's definition of Markov compacta, introduce a subclass of Markov compacta defined by one building block and study cohomological dimensions of these compacta. We show that for a Markov compactum $X$, $\dim_{\Z_{(p)}}X=\dim_{\Q}X$ for all but finitely many primes $p$ where $\Z_{(p)}$ is the localization of $\Z$ at $p$. We construct Markov compacta of arbitrarily large dimension having $\dim_{\Q}X=1$ as well as Markov compacta of arbitrary large rational dimension with $\dim_{\Z_p}X=1$ for a given $p$.