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, dimZ(p)X=dimQX 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 dimQX=1 as well as Markov compacta of arbitrary large rational dimension with dimZpX=1 for a given p.