Geodesic
Profinite groups associated with weakly primitive substitutions
Fundamentalʹnaâ i prikladnaâ matematika, Tome 11 (2005) no. 3, pp. 13-48.

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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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J. Almeida. Profinite groups associated with weakly primitive substitutions. Fundamentalʹnaâ i prikladnaâ matematika, Tome 11 (2005) no. 3, pp. 13-48. http://geodesic.mathdoc.fr/item/FPM_2005_11_3_a1/

[1] Almeida J., “Some algorithms on the star operation applied to finite languages”, Semigroup Forum, 28 (1984), 187–197 | DOI | MR | Zbl

[2] Almeida J., Finite Semigroups and Universal Algebra, World Scientific, Singapore, 1995 | MR | Zbl

[3] Almeida J., “Dynamics of implicit operations and tameness of pseudovarieties of groups”, Trans. Amer. Math. Soc., 354 (2002), 387–411 | DOI | MR | Zbl

[4] Almeida J., “Finite semigroups: An introduction to a unified theory of pseudovarieties”, Semigroups, Algorithms, Automata and Languages, eds. G. M. S. Gomes, J.- E. Pin, P. V. Silva, World Scientific, Singapore, 2002, 3–64 | MR | Zbl

[5] Almeida J., Profinite semigroups and applications, Tech. Rep. CMUP No 2003-33, Univ. Porto, 2003 | MR

[6] Almeida J., “Profinite structures and dynamics”, CIM Bulletin, 14 (2003), 8–18

[7] Almeida J., Symbolic dynamics in free profinite semigroups, No. 1366 in RIMS Kokyuroku, Kyoto, Japan, April 2004, 1–12

[8] Almeida J., Steinberg B., “On the decidability of iterated semidirect products and applications to complexity”, Proc. London Math. Soc., 80 (2000), 50–74 | DOI | MR | Zbl

[9] Almeida J., Steinberg B., “Syntactic and global semigroup theory, a synthesis approach”, Algorithmic Problems in Groups and Semigroups, eds. J. C. Birget, S. W. Margolis, J. Meakin, M. V. Sapir, Birkhäuser, 2000, 1–23 | MR | Zbl

[10] Almeida J., Volkov M. V., “Subword complexity of profinite words and subgroups of free profinite semigroups”, Internat. J. Algebra Comput. (to appear) | DOI | MR

[11] Almeida J., Weil P., “Relatively free profinite monoids: An introduction and examples”, Semigroups, Formal Languages and Groups, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 466, ed. J. B. Fountain, Kluwer Academic, Dordrecht, 1995, 73–117 | MR | Zbl

[12] Berstel J., “Recent results on extensions of Sturmian words”, Internat. J. Algebra Comput., 12 (2002), 371–385 | DOI | MR | Zbl

[13] Berstel J., Perrin D., Theory of Codes, Academic Press, New York, 1985 | MR | Zbl

[14] V. Berthé, S. Ferenczi, C. Mauduit, A. Siegel (eds.), Introduction to Finite Automata and Substitution Dynamical Systems, , 2001 http://iml.univ-mrs.fr/editions/preprint00/book/prebookdac.html

[15] Coulbois T., Sapir M., Weil P., “A note on the continuous extensions of injective morphisms between free groups to relatively free profinite groups”, Publ. Mat., 47 (2003), 477–487 | MR | Zbl

[16] Hall M., “A topology for free groups and related groups”, Ann. Math., 52 (1950), 127–139 | DOI | MR

[17] Lallement G., Semigroups and Combinatorial Applications, Wiley, New York, 1979 | MR | Zbl

[18] Lothaire M., Algebraic Combinatorics on Words, Cambridge University Press, Cambridge, 2002 | MR | Zbl

[19] Margolis S., Sapir M., Weil P., “Irreducibility of certain pseudovarieties”, Comm. Algebra, 26 (1998), 779–792 | DOI | MR | Zbl

[20] Markowsky G., “Bounds on the index and period of a binary relation on a finite set”, Semigroup Forum., 13 (1976), 253–259 | DOI | MR

[21] Queffélec M., Substitution Dynamical Systems. Spectral Analysis, Lect. Notes Math., 1294, Springer, Berlin, 1987 | MR | Zbl

[22] Spehner J.-C., “Quelques constructions et algorithmes relatifs aux sous-mono{\"i}des d'un mono{\"i}de libre”, Semigroup Forum., 9 (1975), 334–353 | DOI | MR

[23] Weil P., “Profinite methods in semigroup theory”, Internat. J. Algebra Comput., 12 (2002), 137–178 | DOI | MR | Zbl

[24] Wielandt H., “Unzerlegbare, nicht negative Matrizen”, Math. Z., 52 (1950), 642–648 | DOI | MR | Zbl