Pronilpotent quotients associated with primitive substitutions (Q2144383)

Let \(\widehat F=\widehat F(A)\) be a free group on a finite base \(A\). Then then \(\mathrm{End}(\widehat F)\), the space of continuous endomorphisms of \(\widehat F\) equipped with composition and the pointwise topology, is a profinite monoid. In particular, for every endomorphism \(\psi \in\mathrm{End}(\widehat F)\), the sequence \((\psi^n)\), \(n\geq 1\) has a unique idempotent accumulation point given by \(\psi^\omega = \lim_n \psi^{n!}\). A profinite group \(G\) is called \(\omega\)-presented when it admits a presentation of the form \(G =\langle A |\hat\psi^{\omega}(a)a^{-1} =1\mid a\in A\rangle\), where A is a finite set and \(\hat \psi\) is an extension of an endomorpism \(\psi\) of a free group \(F(A)\). One then says that \(\psi\) defines an \(\omega\)-presentation of \(G\). Every \(\omega\)-presented group is projective and in the paper [\textit{F. Grunewald} et al., J. Pure Appl. Algebra 218, No. 5, 804--828 (2014; Zbl 1307.20025)] a problem of describing \(\omega\)-presented profinite groups (in slightly different terminology) was posed (see Problem 2 in Section 5). Here, the author describes pronilpotent quotients of such groups; this can be considered as the solution of the above problem in the pronilpotent case. Let \(\varphi\) be an endomorphism of a free group \(F=F(A)\) over a finite base \(A\). Then for each \(p\) it induces an endomorphism \(\varphi_p\) of \(\mathbb{F}_p\)-vector space \(V=F/[F,F]F^p\) whose rank (as linear transformation) we denote by \(\mathfrak m_{p,\varphi}\). Then the main theorem of the author is equivalent to the following statement Theorem. If \(\psi \in \mathrm{End}(F)\) defines an ω-presentation of a profinite group \(G\), then the maximal pronilpotent quotient \(Q_{\mathrm{nil}}(G)=\prod_p \widehat F_p(\mathfrak m_{p,\varphi})\) is a direct product of free pro-\(p\) groups of rank \(\mathfrak m_{p,\varphi}\). Then the author deduces a charcterization of perfect \(\omega\)-presented profinite groups. Namely if one denotes by \(\varphi_{ab}\) the endomorphism induced by \(\varphi\) on the abelianization \(F/[F,F]\) of \(F\), the author proves that \(G\) is perfect if and only if \(\varphi_{ab}\) is nilpotent. As a corollary, the author deduces that either \(G\) is perfect or isomorphic to \(\widehat{\mathbb{Z}}_{\pi}\), where \(\pi\) is the set of almost all primes. The author applies this result to study maximal subgroups of free profinite monoids corresponding to primitive substitutions.