Sets of solutions to systems of equations with finitely many variables in a free group, are equivalent to sets of homomorphisms from a fixed finitely generated group into a free group. The latter can be encoded in a diagram, which is known to be canonical for a fixed finitely generated group with a fixed generating set. In this paper we prove that the construction depends on the chosen generating set of the given finitely generated group.
Des ensembles de solutions à des systèmes d’équations à nombre fini de variables dans un groupe libre, sont équivalents à des ensembles d’homomorphismes d’un groupe de type fini fixé en un groupe libre. Chacun de ces ensembles peut être codé dans un diagramme, qui est connu pour être canonique pour un groupe de type fini fixé avec une partie génératrice fixée. Dans cet article, nous montrons que la construction dépend de la partie génératrice choisie pour le groupe de type fini donné.
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Keywords: Makanin–Razborov diagram, generating set, limit group, JSJ decomposition, Ivanov word, modular automorphism, model theory of groups
Mot clés : diagramme de Makanin–Razborov, partie génératrice, groupe limite, decomposition JSJ, mot Ivanov, automorphisme modulaire, théorie des modèles des groupes
@article{AIF_2020__70_5_2027_0, author = {Berk, Gili}, title = {Canonicality of {Makanin{\textendash}Razborov} {Diagrams} {\textendash} {Counterexample}}, journal = {Annales de l'Institut Fourier}, pages = {2027--2047}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {70}, number = {5}, year = {2020}, doi = {10.5802/aif.3368}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.3368/} }
TY - JOUR AU - Berk, Gili TI - Canonicality of Makanin–Razborov Diagrams – Counterexample JO - Annales de l'Institut Fourier PY - 2020 SP - 2027 EP - 2047 VL - 70 IS - 5 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.3368/ DO - 10.5802/aif.3368 LA - en ID - AIF_2020__70_5_2027_0 ER -
%0 Journal Article %A Berk, Gili %T Canonicality of Makanin–Razborov Diagrams – Counterexample %J Annales de l'Institut Fourier %D 2020 %P 2027-2047 %V 70 %N 5 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.3368/ %R 10.5802/aif.3368 %G en %F AIF_2020__70_5_2027_0
Berk, Gili. Canonicality of Makanin–Razborov Diagrams – Counterexample. Annales de l'Institut Fourier, Volume 70 (2020) no. 5, pp. 2027-2047. doi : 10.5802/aif.3368. http://archive.numdam.org/articles/10.5802/aif.3368/
[1] Algebraic geometry over groups. I: Algebraic sets and ideal theory, J. Algebra, Volume 219 (1999) no. 1, pp. 16-79 | DOI | MR
[2] A combination theorem for negatively curved groups, J. Differ. Geom., Volume 35 (1992) no. 1, pp. 85-101 | DOI | MR
[3] Notes on Sela’s work: limit groups and Makanin–Razborov diagrams, Geometric and cohomological methods in group theory (London Mathematical Society Lecture Note Series), Volume 358, Cambridge University Press, 2009, pp. 1-29 | MR
[4] The isomorphism problem for toral relatively hyperbolic groups, Publ. Math., Inst. Hautes Étud. Sci., Volume 107 (2008) no. 1, pp. 211-290 | DOI | Numdam | MR
[5] JSJ decompositions of doubles of free groups (2018) (https://arxiv.org/abs/1611.01424v2)
[6] On certain elements of free groups, J. Algebra, Volume 204 (1998) no. 2, pp. 394-405 | DOI | MR
[7] Hyperbolic groups and free constructions, Trans. Am. Math. Soc., Volume 350 (1998) no. 2, pp. 571-613 | DOI | MR
[8] On certain C-test words for free groups, J. Algebra, Volume 247 (2002) no. 2, pp. 509-540 | DOI | MR | Zbl
[9] Cyclic splittings of finitely presented groups and the canonical JSJ decomposition, Ann. Math., Volume 146 (1997) no. 1, pp. 53-109 | DOI | MR | Zbl
[10] Structure and rigidity in (Gromov) hyperbolic groups and discrete groups in rank 1 Lie groups II, Geom. Funct. Anal., Volume 7 (1997) no. 3, pp. 561-593 | DOI | MR | Zbl
[11] Diophantine geometry over groups I: Makanin–Razborov diagrams, Publ. Math., Inst. Hautes Étud. Sci., Volume 93 (2001) no. 1, pp. 31-106 | DOI | Numdam | MR | Zbl
[12] An introduction to limit groups (2005) Series for Telgiggy, Imperial College (3-3-05) (cit. on p. 202)
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