Geodesic
Spectral data for a~pair of matrices of order three and an action of the group $\mathrm{GL}(2,\mathbb Z)$
Izvestiya. Mathematics , Tome 75 (2011) no. 5, pp. 959-969.

Voir la notice de l'article provenant de la source Math-Net.Ru

We consider the 10-dimensional complex space whose points are cubic curves on the projective complex plane with three marked points. The triples of marked points on the curve are defined up to equivalence of divisors. We construct a natural action of the group $\mathrm{GL}(2,\mathbb Z)$ on this space.
Keywords: elliptic curve, spectral curve, free group. 
@article{IM2_2011_75_5_a4,
     author = {Yu. A. Neretin},
     title = {Spectral data for a~pair of matrices of order three and an action of the group $\mathrm{GL}(2,\mathbb Z)$},
     journal = {Izvestiya. Mathematics },
     pages = {959--969},
     publisher = {mathdoc},
     volume = {75},
     number = {5},
     year = {2011},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2011_75_5_a4/}
}
TY  - JOUR
AU  - Yu. A. Neretin
TI  - Spectral data for a~pair of matrices of order three and an action of the group $\mathrm{GL}(2,\mathbb Z)$
JO  - Izvestiya. Mathematics 
PY  - 2011
SP  - 959
EP  - 969
VL  - 75
IS  - 5
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_2011_75_5_a4/
LA  - en
ID  - IM2_2011_75_5_a4
ER  - 
%0 Journal Article
%A Yu. A. Neretin
%T Spectral data for a~pair of matrices of order three and an action of the group $\mathrm{GL}(2,\mathbb Z)$
%J Izvestiya. Mathematics 
%D 2011
%P 959-969
%V 75
%N 5
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_2011_75_5_a4/
%G en
%F IM2_2011_75_5_a4
Yu. A. Neretin. Spectral data for a~pair of matrices of order three and an action of the group $\mathrm{GL}(2,\mathbb Z)$. Izvestiya. Mathematics , Tome 75 (2011) no. 5, pp. 959-969. http://geodesic.mathdoc.fr/item/IM2_2011_75_5_a4/

[1] R. C. Penner, “The decorated Teichmüller space of punctured surfaces”, Comm. Math. Phys., 113:2 (1988), 299–339 | DOI | MR | Zbl

[2] A. A. Klyachko, “Spatial polygons and stable configurations of points in the projective line”, Algebraic geometry and its applications (Yaroslavl', Russia, 1992), Aspects Math., 25, Vieweg, Braunschweig, 1994, 67–84 | MR | Zbl

[3] V. V. Fock, Dual Teichmüller space, arXiv: dg-ga/9702018

[4] L. O. Chekhov, V. V. Fock, “A quantum Teichmüller space”, Theoret. and Math. Phys., 120:3 (1999), 1245–1259 | DOI | MR | Zbl

[5] V. V. Fock, A. B. Goncharov, “Dual Teichmüller and lamination spaces”, Handbook of Teichmüller theory, v. 1, IRMA Lect. Math. Theor. Phys., 11, Eur. Math. Soc., Zürich, 2007, 647–684 | MR | Zbl

[6] Yu. A. Neretin, “On spherical functions on the group $\mathrm{SU}(2)\times \mathrm{SU}(2) \times \mathrm{SU}(2)$”, Funct. Anal. Appl., 44:1 (2010), 48–54 | DOI | MR

[7] Yu. A. Neretin, “Double cosets for $\mathrm{SU}(2)\times \dots\times\mathrm{SU}(2)$ and outer automorphisms of free groups”, Int. Math. Res. Notices, 2011, no. 9, 2047–2067 | DOI | Zbl

[8] Yu. A. Neretin, Infinite tri-symmetric group, multiplication of double cosets, and checker topological field theories, arXiv: 0909.4739

[9] Yu. A. Neretin, Categories of symmetries and infinite-dimensional groups, London Math. Soc. Monogr. (N.S.), 16, Clarendon Press, Oxford, 1996 | MR | Zbl

[10] N. I. Nessonov, “Faktor-predstavleniya gruppy $\mathrm{GL}(\infty)$ i dopustimye predstavleniya gruppy $\mathrm{GL}(\infty)^X$”, Matem. fizika, analiz, geom. Kharkovskii matem. zhurn., 10:2 (2003), 167–178 | MR | Zbl

[11] N. I. Nessonov, “Faktorpredstavleniya gruppy $\mathrm{GL}(\infty)$ i dopustimye predstavleniya gruppy $\mathrm{GL}(\infty)^X$. II”, Matem. fizika, analiz, geom. Kharkovskii matem. zhurn., 10:4 (2003), 524–556 | MR | Zbl

[12] Yu. A. Neretin, Multi-operator colligations and multivariate characteristic functions, arXiv: 1006.2275

[13] Yu. A. Neretin, Sphericity and multiplication of double cosets for infinite-dimensional classical groups, arXiv: 1101.4759

[14] B. Farb, D. Margalit, A primer of mapping class groups, http://www.math.utah.edu/ margalit/primer

[15] N. Hitchin, “Riemann surfaces and integrable systems”, Integrable systems (Oxford, UK, 1997), Oxf. Grad. Texts Math., 4, Clarendon Press, Oxford, 1999, 11–52 | MR | Zbl

[16] V. V. Fock, A. A. Rosly, “Flat connections and polyubles”, Theor. Math. Phys, 95:2 (1993), 526–534 | DOI | MR | Zbl

[17] V. V. Fock, A. A. Rosly, “Poisson structure on moduli of flat connections on Riemann surfaces and the $r$-matrix”, Moscow Seminar in mathematical physics, Amer. Math. Soc. Transl. Ser. 2, 191, Providence, RI, 1999, 67–86 | MR | Zbl

[18] R. J. Cook, A. D. Thomas, “Line bundles and homogeneous matrices”, Quart. J. Math. Oxford Ser. (2), 30:4 (1979), 423–429 | DOI | MR | Zbl

[19] V. Vinnikov, “Determinantal representations of algebraic curves”, Linear algebra in signals, systems, and control (Boston, MA, 1986), SIAM, Philadelphia, PA, 1988, 73–99 | MR | Zbl

[20] A. Beauville, “Determinantal hypersurfaces”, Michigan Math. J., 48:1 (2000), 39–64 | DOI | MR | Zbl

[21] A. N. Tyurin, “On intersections of quadrics”, Russian Math. Surveys, 30:6 (1975), 51–105 | DOI | MR | Zbl | Zbl

[22] M. G. Reiman, M. S. Semenov-Tyan-Shanskii, Integriruemye sistemy. Teoretiko-gruppovoi podkhod, In-t kompyuternykh issledovanii, M.–Izhevsk, 2003

[23] R. C. Lyndon, P. E. Schupp, Combinatorial group theory, Springer-Verlag, Berlin–Heidelberg–New York, 1977 | MR | MR | Zbl | Zbl

[24] Ph. Griffiths, J. Harris, Principles of algebraic geometry, Wiley, New York, 1978 | MR | MR | Zbl | Zbl

[25] C. H. Clemens, A scrapbook of complex curve theory, Plenum, New York–London, 1980 | MR | MR | Zbl | Zbl