Dynamics on Character Varieties and Malgrange irreducibility of Painlevé VI equation

Cantat, Serge; Loray, Frank

Annales de l'Institut Fourier, Tome 59 (2009) no. 7, pp. 2927-2978.

Nous étudions l’action du groupe modulaire sur l’espace des représentations du groupe fondamental de la sphère privée de quatre points dans $\mathrm{SL}(2,ℂ)$. Ce système dynamique peut être interprété comme la monodromie de l’équation de Painlevé VI. Nous caractérisons les orbites infinies bornées : elles proviennent des représentations dans $\mathsf{SU}\left(\mathsf{2}\right)$. Nous démontrons l’absence de struture affine invariante (excepté pour des paramètres spéciaux) puis déduisons, en nous appuyant sur des travaux de Casale, que le groupoïde de Malgrange associé est le groupoïde symplectique. Ceci permet de donner une preuve de l’irréductibilité de l’équation de Painlevé VI, c’est-à-dire de la forte transcendance de ses solutions, par une approche galoisienne, dans l’esprit de la tentative de Drach et Painlevé.

We consider representations of the fundamental group of the four punctured sphere into $\mathrm{SL}(2,ℂ)$. The moduli space of representations modulo conjugacy is the character variety. The Mapping Class Group of the punctured sphere acts on this space by symplectic polynomial automorphisms. This dynamical system can be interpreted as the monodromy of the Painlevé VI equation. Infinite bounded orbits are characterized: they come from $\mathsf{SU}\left(\mathsf{2}\right)$-representations. We prove the absence of invariant affine structure (and invariant foliation) for this dynamical system except for special explicit parameters. Following results of Casale, this implies that Malgrange’s groupoid of the Painlevé VI foliation coincides with the symplectic one. This provides a new proof of the transcendence of Painlevé solutions.

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