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Endomorphisms and bijections of the character variety χ(F 2 ,SL 2 (C))
Serge Cantat1
1 Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes (France)
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 29 (2020) no. 4, pp. 897-906.

We answer a question of Gelander and Souto in the special case of the free group of rank 2. The result may be stated as follows. If F is a free group of rank 2, and G is a proper subgroup of F, the restriction of homomorphisms F→SL 2 (ℂ) to the subgroup G defines a map from the character variety χ(F,SL 2 (C)) to the character variety χ(G,SL 2 (C)); this algebraic map never induces a bijection between these two character varieties.

Le résultat suivant, qui répond à une question de Gelander et Souto dans un cas particulier, est démontré : si F est le groupe libre de rang 2 et G est un sous-groupe de F, la restriction des homomorphismes F→SL 2 (C) au sous-groupe G fournit une application de la variété des caractères χ(F,SL 2 (C)) vers la variété des caractères χ(G,SL 2 (C)) ; cette application algébrique n’est bijective que si G coïncide avec F.

Received:
Accepted:
Published online:
DOI: 10.5802/afst.1648

Serge Cantat 1

1 Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes (France)
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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     title = {Endomorphisms and bijections of the character variety $\chi (\protect \mathbf{F}_2,\protect \mathsf {SL}_2(\protect \mathbf{C}))$},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
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Serge Cantat. Endomorphisms and bijections of the character variety $\chi (\protect \mathbf{F}_2,\protect \mathsf {SL}_2(\protect \mathbf{C}))$. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 29 (2020) no. 4, pp. 897-906. doi : 10.5802/afst.1648. https://afst.centre-mersenne.org/articles/10.5802/afst.1648/

[1] Serge Cantat Bers and Hénon, Painlevé and Schroedinger, Duke Math. J., Volume 149 (2009) no. 3, pp. 411-460 | DOI | MR | Zbl

[2] Serge Cantat; Frank Loray Dynamics on character varieties and Malgrange irreducibility of Painlevé VI equation, Ann. Inst. Fourier, Volume 59 (2009) no. 7, pp. 2927-2978 | DOI | Numdam | Zbl

[3] Marat H. Èl’-Huti Cubic surfaces of Markov type, Mat. Sb., N. Ser., Volume 93(135) (1974), p. 331-346, 487

[4] Arno van den Essen Polynomial Automorphisms, and the Jacobian Conjecture, Progress in Mathematics, 190, Birkhäuser, 2000 | MR | Zbl

[5] William M. Goldman Trace coordinates on Fricke spaces of some simple hyperbolic surfaces, Handbook of Teichmüller theory. Vol. II (IRMA Lectures in Mathematics and Theoretical Physics), Volume 13, European Mathematical Society, 2009, pp. 611-684 | DOI | MR | Zbl

[6] Pierre de la Harpe Topics in geometric group theory, Chicago Lectures in Mathematics, University of Chicago Press, 2000, vi+310 pages | Zbl

[7] Roger C. Lyndon; Paul E. Schupp Combinatorial group theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, 89, Springer, 1977, xiv+339 pages | MR | Zbl

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