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Endomorphisms and bijections of the character variety χ(F 2 ,SL 2 (C))
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 FSL 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 FSL 2 () 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.

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DOI : https://doi.org/10.5802/afst.1648
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     author = {Serge Cantat},
     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},
     pages = {897--906},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 29},
     number = {4},
     year = {2020},
     doi = {10.5802/afst.1648},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1648/}
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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, Série 6, Tome 29 (2020) no. 4, pp. 897-906. doi : 10.5802/afst.1648. https://afst.centre-mersenne.org/articles/10.5802/afst.1648/

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