Pronilpotent quotients associated with primitive substitutions

TL;DR

This paper investigates the pronilpotent quotients of a special class of profinite groups called $\,\omega$-presented groups, revealing their dependence on a polynomial and implications for group freeness, motivated by semigroup theory.

Contribution

It introduces the concept of $\,\omega$-presented groups, characterizes their pronilpotent quotients via a polynomial, and links these findings to properties of maximal subgroups in free profinite monoids.

Findings

Pronilpotent quotients are determined by a characteristic polynomial.

$\,\omega$-presented groups are either perfect or have $p$-adic integer quotients.

Maximal subgroups of primitive aperiodic substitutions are not absolutely free.

Abstract

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.