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 Fundam. Prikl. Mat., 2005, Volume 11, Issue 3, Pages 13–48 (Mi fpm826)

Profinite groups associated with weakly primitive substitutions

J. Almeida

University of Porto

Abstract: A uniformly recurrent pseudoword is an element of a free profinite semigroup in which every finite factor appears in every sufficiently long finite factor. An alternative characterization is as a pseudoword that is a factor of all its infinite factors, i.e., one that lies in a $\mathcal J$-class with only finite words strictly $\mathcal J$-above it. Such a $\mathcal J$-class is regular, and therefore it has an associated profinite group, namely any of its maximal subgroups. One way to produce such $\mathcal J$-classes is to iterate finite weakly primitive substitutions. This paper is a contribution to the computation of the profinite group associated with the $\mathcal J$-class that is generated by the infinite iteration of a finite weakly primitive substitution. The main result implies that the group is a free profinite group provided the substitution induced on the free group on the letters that appear in the images of all of its sufficiently long iterates is invertible.

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English version:
Journal of Mathematical Sciences (New York), 2007, 144:2, 3881–3903

Bibliographic databases:

UDC: 512.53

Citation: J. Almeida, “Profinite groups associated with weakly primitive substitutions”, Fundam. Prikl. Mat., 11:3 (2005), 13–48; J. Math. Sci., 144:2 (2007), 3881–3903

Citation in format AMSBIB
\Bibitem{Alm05} \by J.~Almeida \paper Profinite groups associated with weakly primitive substitutions \jour Fundam. Prikl. Mat. \yr 2005 \vol 11 \issue 3 \pages 13--48 \mathnet{http://mi.mathnet.ru/fpm826} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2176678} \zmath{https://zbmath.org/?q=an:1110.20022} \transl \jour J. Math. Sci. \yr 2007 \vol 144 \issue 2 \pages 3881--3903 \crossref{https://doi.org/10.1007/s10958-007-0242-y} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-34250161770} 

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This publication is cited in the following articles:
1. Rhodes J., Steinberg B., “Closed subgroups of free profinite monoids are projective profinite groups”, Bull. Lond. Math. Soc., 40:3 (2008), 375–383
2. Almeida J., Steinberg B., “Rational codes and free profinite monoids”, J. Lond. Math. Soc. (2), 79:2 (2009), 465–477
3. Almeida J., Costa A., “Infinite-vertex free profinite semigroupoids and symbolic dynamics”, J. Pure Appl. Algebra, 213:5 (2009), 605–631
4. Steinberg B., “Maximal subgroups of the minimal ideal of a free profinite monoid are free”, Israel J. Math., 176:1 (2010), 139–155
5. Costa A., Steinberg B., “Profinite groups associated to sofic shifts are free”, Proc. Lond. Math. Soc., 102:2 (2011), 341–369
6. Steinberg B., “On the endomorphism monoid of a profinite semigroup”, Portugaliae Mathematica, 68:2 (2011), 177–183
7. Almeida J., Costa A., “On the Transition Semigroups of Centrally Labeled Rauzy Graphs”, Internat J Algebra Comput, 22:2 (2012), 1250018
8. Almeida J., Costa A., “Presentations of Schutzenberger Groups of Minimal Subshifts”, Isr. J. Math., 196:1 (2013), 1–31
9. Almeida J., Costa J.C., Zeitoun M., “Iterated Periodicity Over Finite Aperiodic Semigroups”, Eur. J. Comb., 37:SI (2014), 115–149
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