The parallelogram identity on groups and deformations of the trivial character in ${\mathrm{SL}}_2\left(ℂ\right)$

Marché, Julien; Wolff, Maxime

Journal de l’École polytechnique — Mathématiques, Tome 7 (2020), pp. 263-285.

On décrit sur tout groupe de type fini $\Gamma$ l’espace de toutes les fonctions $f:\Gamma \to ℂ$ qui satisfont à l’identité du parallélogramme, $f\left(xy\right)+f\left(x{y}^{-1}\right)=2f\left(x\right)+2f\left(y\right)$. Il est connu (mais peu) que ces fonctions correspondent aux vecteurs Zariski-tangents au caractère trivial dans la variété des caractères de $\Gamma$ dans ${\mathrm{SL}}_2\left(ℂ\right)$. 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 $dim{H}_1(\Gamma ,ℂ)<2$ et singulier si $dim{H}_1(\Gamma ,ℂ)>2$.

We describe on any finitely generated group $\Gamma$ the space of maps $\Gamma \to ℂ$ which satisfy the parallelogram identity, $f\left(xy\right)+f\left(x{y}^{-1}\right)=2f\left(x\right)+2f\left(y\right)$. It is known (but not well-known) that these functions correspond to Zariski-tangent vectors at the trivial character of the character variety of $\Gamma$ in ${\mathrm{SL}}_2\left(ℂ\right)$. 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 $dim{H}_1(\Gamma ,ℂ)<2$ and not a smooth point if $dim{H}_1(\Gamma ,ℂ)>2$.

Classification : 20F14, 20G05, 20J05, 14B05, 14L24
Keywords: Character variety, group homology, deformation theory, polynomial functions on groups
Mot clés : Variétés de caractères, homologie des groupes, théorie de la déformation, fonctions polynomiales sur les groupes

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