Modified hypergeometric equations arising from the Markoff theory

Abstract:
After recalling what is the Markoff theory, the article summarizes some links which exist with the group GL(2,\mathbbZ) of 2×2 matrices with integer coefficients and determinant ±1 and with its subgroups SL(2,\mathbbZ) and the triangle group T₃. Then we visit rapidly the links with conformal punctured toruses. The main part of the article is about the monodromy representation of the Poincaré group of such a torus. We gives the corresponding solution of the associated Riemann-Hilbert problem and the corresponding differential operator whose spectral analysis remains to be done. We conclude quoting the 22^th Hilbert's problem and some information about the accessory parameter problem.
 

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