The parallelogram identity on groups and deformations of the trivial character in SL 2 ()
Journal de l’École polytechnique — Mathématiques, Volume 7 (2020), pp. 263-285.

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.

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.

Received:
Accepted:
Published online:
DOI: 10.5802/jep.117
Classification: 20F14,  20G05,  20J05,  14B05,  14L24
Keywords: Character variety, group homology, deformation theory, polynomial functions on groups
Julien Marché 1; Maxime Wolff 1

1 Sorbonne Université, Université Paris Diderot, CNRS, Institut de Mathématiques de Jussieu-Paris Rive Gauche, IMJ-PRG F-75005, Paris, France
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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, Volume 7 (2020), pp. 263-285. doi : 10.5802/jep.117. https://jep.centre-mersenne.org/articles/10.5802/jep.117/

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