The parallelogram identity on groups and deformations of the trivial character in SL 2 ()
[L’identité du parallélogramme sur les groupes et les déformations du caractère trivial dans $\protect \mathrm{SL}_2(\protect \mathbb{C})$]
Journal de l'École polytechnique — Mathématiques, Tome 7 (2020) , pp. 263-285.

On décrit sur tout groupe de type fini Γ l’espace de toutes les fonctions f:Γ qui satisfont à l’identité du parallélogramme, f(xy)+f(xy -1 )=2f(x)+2f(y). Il est connu (mais peu) que ces fonctions correspondent aux vecteurs Zariski-tangents au caractère trivial dans la variété des caractères de Γ dans SL 2 (). On étudie les obstructions à déformer le caractère trivial dans la direction donnée par f. Au passage, on montre que le caractère trivial est lisse si dimH 1 (Γ,)<2 et singulier si dimH 1 (Γ,)>2.

We describe on any finitely generated group Γ the space of maps Γ which satisfy the parallelogram identity, f(xy)+f(xy -1 )=2f(x)+2f(y). It is known (but not well-known) that these functions correspond to Zariski-tangent vectors at the trivial character of the character variety of Γ in SL 2 (). We study the obstructions for deforming the trivial character in the direction given by f. Along the way, we show that the trivial character is a smooth point of the character variety if dimH 1 (Γ,)<2 and not a smooth point if dimH 1 (Γ,)>2.

Reçu le : 2018-09-04
Accepté le : 2020-02-04
Publié le : 2020-02-13
DOI : https://doi.org/10.5802/jep.117
Classification : 20F14,  20G05,  20J05,  14B05,  14L24
Mots clés: Variétés de caractères, homologie des groupes, théorie de la déformation, fonctions polynomiales sur les groupes
@article{JEP_2020__7__263_0,
     author = {Julien March\'e and Maxime Wolff},
     title = {The parallelogram identity on groups and deformations of the trivial character in~$\protect \mathrm{SL}\_2(\protect \mathbb{C})$},
     journal = {Journal de l'\'Ecole polytechnique --- Math\'ematiques},
     publisher = {\'Ecole polytechnique},
     volume = {7},
     year = {2020},
     pages = {263-285},
     doi = {10.5802/jep.117},
     zbl = {07168067},
     language = {en},
     url = {jep.centre-mersenne.org/item/JEP_2020__7__263_0/}
}
Julien Marché; Maxime Wolff. The parallelogram identity on groups and deformations of the trivial character in $\protect \mathrm{SL}_2(\protect \mathbb{C})$. Journal de l'École polytechnique — Mathématiques, Tome 7 (2020) , pp. 263-285. doi : 10.5802/jep.117. https://jep.centre-mersenne.org/item/JEP_2020__7__263_0/

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