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

Julien Marché; Maxime Wolff

doi:10.5802/jep.117

The parallelogram identity on groups and deformations of the trivial character in

{SL}_{2} (ℂ)

[L’identité du parallélogramme sur les groupes et les déformations du caractère trivial dans

{SL}_{2} (ℂ)

]

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

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

Résumé
Abstract

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

We describe on any finitely generated group $Γ$ the space of maps $Γ \to ℂ$ which satisfy the parallelogram identity, $f (x y) + f (x y^{- 1}) = 2 f (x) + 2 f (y)$ . 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 ${SL}_{2} (ℂ)$ . 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 $dim H_{1} (Γ, ℂ) < 2$ and not a smooth point if $dim H_{1} (Γ, ℂ) > 2$ .

Reçu le : 2018-09-04
Accepté le : 2020-02-04
Publié le : 2020-02-13

Zbl

DOI : 10.5802/jep.117

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

Affiliations des auteurs :

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

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {The parallelogram identity on groups and deformations of the trivial character in~$\protect \mathrm{SL}_2(\protect \mathbb{C})$},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {263--285},
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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, Tome 7 (2020), pp. 263-285. doi : 10.5802/jep.117. https://jep.centre-mersenne.org/articles/10.5802/jep.117/

Bibliographie
Cité par

[1] M. Artin - “On the solutions of analytic equations”, Invent. Math. 5 (1968), p. 277-291 | DOI

[2] M. Artin - “On Azumaya algebras and finite dimensional representations of rings”, J. Algebra 11 (1969), p. 532-563 | DOI | MR | Zbl

[3] K. S. Brown - Cohomology of groups, Graduate Texts in Math., vol. 87, Springer-Verlag, New York, 1994

[4] G. W. Brumfiel & H. M. Hilden - $SL (2)$ representations of finitely presented groups, Contemporary Math., vol. 187, American Mathematical Society, Providence, RI, 1995 | DOI | MR | Zbl

[5] G. Chenevier - “Sur la variété des caractères $p$ -adique du groupe de Galois absolu de $ℚ_{p}$ ”, 2009

[6] G. Chenevier - “Mémoire de HDR: Représentations galoisiennes automorphes et conséquences arithmétiques des conjectures de Langlands et Arthur”, 2013, http://gaetan.chenevier.perso.math.cnrs.fr/hdr/HDR.pdf

[7] M. Culler & P. B. Shalen - “Varieties of group representations and splittings of $3$ -manifolds”, Ann. of Math. (2) 117 (1983) no. 1, p. 109-146 | DOI | MR | Zbl

[8] D. B. A. Epstein - “Finite presentations of groups and $3$ -manifolds”, Quart. J. Math. Oxford Ser. (2) 12 (1961), p. 205-212 | DOI | MR | Zbl

[9] B. Farb & D. Margalit - A primer on mapping class groups, Princeton Math. Series, vol. 49, Princeton University Press, Princeton, NJ, 2012 | MR | Zbl

[10] L. Funar & J. Marché - “The first Johnson subgroups act ergodically on ${SU}_{2}$ -character varieties”, J. Differential Geom. 95 (2013) no. 3, p. 407-418 | DOI | MR | Zbl

[11] W. M. Goldman - “Mapping class group dynamics on surface group representations”, in Problems on mapping class groups and related topics, Proc. Sympos. Pure Math., vol. 74, American Mathematical Society, Providence, RI, 2006, p. 189-214 | DOI | MR | Zbl

[12] J. H. Grace & A. Young - The algebra of invariants, Cambridge Library Collection, Cambridge University Press, Cambridge, 1903 | DOI | Zbl

[13] Q. Liu - Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Math., vol. 6, Oxford University Press, Oxford, 2002 | MR | Zbl

[14] C. Maclachlan & A. W. Reid - The arithmetic of hyperbolic 3-manifolds, Graduate Texts in Math., vol. 219, Springer-Verlag, New York, 2003 | DOI | MR | Zbl

[15] S. Mukai - An introduction to invariants and moduli, Cambridge Studies in Advanced Math., vol. 81, Cambridge University Press, Cambridge, 2003 | MR | Zbl

[16] D. Mumford, J. Fogarty & F. Kirwan - Geometric invariant theory, Ergeb. Math. Grenzgeb. (2), vol. 34, Springer-Verlag, Berlin, 1994 | DOI | MR | Zbl

[17] A. Nijenhuis & R. W. Richardson - “Cohomology and deformations in graded Lie algebras”, Bull. Amer. Math. Soc. 72 (1966), p. 1-29 | DOI | MR | Zbl

[18] I. B. S. Passi - Group rings and their augmentation ideals, Lect. Notes in Math., vol. 715, Springer, Berlin, 1979 | MR | Zbl

[19] P. de Place Friis & H. Stetkær - “On the quadratic functional equation on groups”, Publ. Math. Debrecen 69 (2006) no. 1-2, p. 65-93 | MR | Zbl

[20] C. Procesi - “The invariant theory of $n \times n$ matrices”, Advances in Math. 19 (1976) no. 3, p. 306-381 | DOI | Zbl

[21] R. Rouquier - “Caractérisation des caractères et pseudo-caractères”, J. Algebra 180 (1996) no. 2, p. 571-586 | DOI | MR | Zbl

[22] K. Saito - “Character variety of representations of a finitely generated group in ${SL}_{2}$ ”, in Topology and Teichmüller spaces (Katinkulta, 1995), World Sci. Publ., River Edge, NJ, 1996, p. 253-264 | DOI | MR | Zbl

[23] J.-P. Serre - Arbres, amalgames, ${SL}_{2}$ , Astérisque, vol. 46, Société Mathématique de France, Paris, 1977 | MR | Zbl

[24] J. Stallings - “Homology and central series of groups”, J. Algebra 2 (1965), p. 170-181 | DOI | MR | Zbl

[25] A. Weil - “Remarks on the cohomology of groups”, Ann. of Math. (2) 80 (1964), p. 149-157 | DOI | MR | Zbl

Cité par Sources :