  We describe the pronilpotent quotients of a class of projective profinite
groups, that we call $\omega$-presented groups, defined using a special type of
presentations. The pronilpotent quotients of an $\omega$-presented group are
completely determined by a single polynomial, closely related with the
characteristic polynomial of a matrix. We deduce that $\omega$-presented groups
are either perfect or admit the $p$-adic integers as quotients for cofinitely
many primes. We also find necessary conditions for absolute and relative
freeness of $\omega$-presented groups. Our main motivation comes from semigroup
theory: the maximal subgroups of free profinite monoids corresponding to
primitive substitutions are $\omega$-presented (a theorem due to Almeida and
Costa). We are able to show that the incidence matrix of a primitive
substitution carries partial information on the pronilpotent quotients of the
corresponding maximal subgroup. We apply this to deduce that the maximal
subgroups corresponding to primitive aperiodic substitutions of constant length
are not absolutely free.
