Résumé | (joint work with Alfredo Costa)
It is well known that the words that appear as factors in the
iteration on the letters of a primitive endomorphism (substitution) f
of the free semigroup on a finite alphabet A is the language of blocks
of a minimal symbolic dynamical system (subshift) X_f, consisting of
biinfinite words over the alphabet A whose blocks are those factors. I
proved in 2005 that, associating to a minimal subshift X over the
alphabet A the closure J(X) of its language of blocks in the profinite
semigroup freely generated by A, one obtains a bijection between
minimal subshifts and J-maximal regular J-classes. My co-author showed
in his thesis that, viewed as an abstract group G(X), the maximal
subgroups of J(X) constitute a conjugacy invariant. I also showed
that, if f induces an automorphism of the free group on A, then G(X_f)
is a free profinite group, while there are examples for which G(X_f),
which is always finitely generated, is not a free profinite group.
Rhodes and Steinberg proved that the closed subgroups of a free
profinite semigroup are projective as profinite groups. Hence, as
observed by Lubotzky, if finitely generated, such groups admit finite
presentations, as profinite groups, in which the relations simply
state that each generator is a fixed point of a retraction of the free
profinite group. I conjectured in 2005 that, under special conditions
on the primitive substitution f, the group G(X_f) admits such a
presentation in which the retraction is obtained as a (profinite)
idempotent iterate of a positive finite continuous endomorphism f' of
the free profinite group, where f' can be effectively computed from f.
The interest in such presentations stems from the fact that the
relations can be effectively checked in a given finite group, so that
the group with such a retract presentation has decidable finite
quotients.
It turns out that the conjecture holds for every primitive
substitution f. It is therefore decidable whether a finite group is a
quotient of G(X_f). The proof of the conjecture in such a wide setting
depends on a synchronization result of Mossé for (biinfinite) fixed
points of primitive substitutions. As an application, we show that the
group associated with the classical Prouhet-Thue-Morse substitution
(a -> ab, b -> ba) is not free. |