On the group property recognition problem

The unsolvability of the problem of deciding whether a class of finitely presented groups in a $(p+3)$-letter alphabet has Markov group properties is proved ($p$ is the number of generators of the finitely presented group having a particular property of the kind in question). The problem of deciding whether a class of finitely presented groups in the minimal $(p+1)$-letter alphabet has Markov properties such that a group having those properties contains an infinite cyclic subgroup is proved to be unsolvable.