no. 4
Endomorphisms and bijections of the character variety χ(F 2 ,SL 2 (C))
Cantat, Serge1
1 Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes (France)
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 29 (2020) no. 4, pp. 897-906.

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.

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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/afst.1648
Cantat, Serge 1

1 Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes (France) 
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Cantat, Serge. 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, Série 6, Tome 29 (2020) no. 4, pp. 897-906. doi : 10.5802/afst.1648. http://www.numdam.org/articles/10.5802/afst.1648/

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

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

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

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

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

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