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Modified hypergeometric equations arising from the Markoff theory

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 22th Hilbert's problem and some information about the accessory parameter problem.

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