Cohomological dimension of Markov compacta

We rephrase Gromovʹs 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.