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Measure-theoretic rigidity for Mumford curves

Published online by Cambridge University Press:  17 April 2012

GUNTHER CORNELISSEN and
JANNE KOOL
GUNTHER CORNELISSEN
Affiliation:
Mathematisch Instituut, Universiteit Utrecht, Postbus 80.010, 3508 TA Utrecht, Nederland (email: g.cornelissen@uu.nl, j.kool2@uu.nl)
JANNE KOOL
Affiliation:
Mathematisch Instituut, Universiteit Utrecht, Postbus 80.010, 3508 TA Utrecht, Nederland (email: g.cornelissen@uu.nl, j.kool2@uu.nl)
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Abstract

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One can describe isomorphism of two compact hyperbolic Riemann surfaces of the same genus by a measure-theoretic property: a chosen isomorphism of their fundamental groups corresponds to a homeomorphism on the boundary of the Poincaré disc that is absolutely continuous for Lebesgue measure if and only if the surfaces are isomorphic. In this paper, we find the corresponding statement for Mumford curves, a non-Archimedean analogue of Riemann surfaces. In this case, the mere absolute continuity of the boundary map (for Schottky uniformization and the corresponding Patterson–Sullivan measure) only implies isomorphism of the special fibers of the Mumford curves, and the absolute continuity needs to be enhanced by a finite list of conditions on the harmonic measures on the boundary (certain non-Archimedean distributions constructed by Schneider and Teitelbaum) to guarantee an isomorphism of the Mumford curves. The proof combines a generalization of a rigidity theorem for trees due to Coornaert, the existence of a boundary map by a method of Floyd, with a classical theorem of Babbage, Enriques and Petri on equations for the canonical embedding of a curve.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 33 , Issue 3 , June 2013, pp. 851 - 869
Copyright
Copyright © 2012 Cambridge University Press

References

[1]Arbarello, E., Cornalba, M., Griffiths, P. A. and Harris, J.. Geometry of Algebraic Curves. Vol. I (Grundlehren der Mathematischen Wissenschaften, 267). Springer, New York, 1985.Google Scholar
[2]Babbage, D. W.. A note on the quadrics through a canonical curve. J. Lond. Math. Soc. 14 (1939), 310–315.CrossRefGoogle Scholar
[3]Baker, M.. Specialization of linear systems from curves to graphs. Algebra Number Theory 2(6) (2008), 613–653.CrossRefGoogle Scholar
[4]Bowen, R.. Hausdorff dimension of quasicircles. Inst. Hautes Études Sci. Publ. Math. (50) (1979), 11–25.Google Scholar
[5]Coornaert, M.. Rigidité ergodique de groupes d’isométries d’arbres. C. R. Acad. Sci. Paris Sér. I Math. 315(3) (1992), 301–304.Google Scholar
[6]Coornaert, M.. Mesures de Patterson–Sullivan sur le bord d’un espace hyperbolique au sens de Gromov. Pacific J. Math. 159(2) (1993), 241–270.CrossRefGoogle Scholar
[7]Coornaert, M., Delzant, T. and Papadopoulos, A.. Géométrie et théorie des groupes (Lecture Notes in Mathematics, 1441). Springer, Berlin, 1990.CrossRefGoogle Scholar
[8]Cornelissen, G., Kato, F. and Kontogeorgis, A.. The relation between rigid-analytic and algebraic deformation parameters for Artin–Schreier–Mumford curves. Israel J. Math. 180 (2010), 345–370.CrossRefGoogle Scholar
[9]Dodane, O.. Théorèmes de Petri pour les courbes stables et dégénérescence du système d’équations du plongement canonique. Université de Strasbourg Thesis, url: http://tel.archives-ouvertes.fr/docs/00/40/42/57/PDF/these-OD.pdf, 2009.Google Scholar
[10]Floyd, W. J.. Group completions and limit sets of Kleinian groups. Invent. Math. 57(3) (1980), 205–218.Google Scholar
[11]Gerritzen, L. and van der Put, M.. Schottky Groups and Mumford Curves (Lecture Notes in Mathematics, 817). Springer, Berlin, 1980.CrossRefGoogle Scholar
[12]Hersonsky, S. and Paulin, F.. On the rigidity of discrete isometry groups of negatively curved spaces. Comment. Math. Helv. 72(3) (1997), 349–388.Google Scholar
[13]Kuusalo, T.. Boundary mappings of geometric isomorphisms of Fuchsian groups. Ann. Acad. Sci. Fenn. Ser. A I(545) (1973), 7.Google Scholar
[14]Lubotzky, A.. Lattices in rank one Lie groups over local fields. Geom. Funct. Anal. 1(4) (1991), 406–431.Google Scholar
[15]Manin, Y. I.. Three-dimensional hyperbolic geometry as $\infty $-adic Arakelov geometry. Invent. Math. 104(2) (1991), 223–243.Google Scholar
[16]Manin, Y. I. and Marcolli, M.. Holography principle and arithmetic of algebraic curves. Adv. Theor. Math. Phys. 5(3) (2001), 617–650.Google Scholar
[17]Mostow, G. D.. Strong Rigidity of Locally Symmetric Spaces (Annals of Mathematics Studies, 78). Princeton University Press, Princeton, NJ, 1973.Google Scholar
[18]Mumford, D.. An analytic construction of degenerating curves over complete local rings. Compositio Math. 24 (1972), 129–174.Google Scholar
[19]Patterson, S. J.. The limit set of a Fuchsian group. Acta Math. 136(3–4) (1976), 241–273.CrossRefGoogle Scholar
[20]Petri, K.. Über die invariante Darstellung algebraischer Funktionen einer Veränderlichen. Math. Ann. 88 (1923), 242–289.CrossRefGoogle Scholar
[21]Roney-Dougal, C. M.. Conjugacy of subgroups of the general linear group. Experiment. Math. 13(2) (2004), 151–163.CrossRefGoogle Scholar
[22]Saint-Donat, B.. On Petri’s analysis of the linear system of quadrics through a canonical curve. Math. Ann. 206 (1973), 157–175.Google Scholar
[23]Schneider, P.. Rigid-analytic $L$-transforms. Number Theory (Noordwijkerhout, 1983) (Lecture Notes in Mathematics, 1068). Springer, Berlin, 1984, pp. 216–230.Google Scholar
[24]Serre, J.-P.. Trees (Springer Monographs in Mathematics). Springer, Berlin, 2003.Google Scholar
[25]Sullivan, D.. The density at infinity of a discrete group of hyperbolic motions. Publ. Math. Inst. Hautes Études Sci. (50) (1979), 171–202.Google Scholar
[26]Tate, J.. Rigid analytic spaces. Invent. Math. 12 (1971), 257–289.Google Scholar
[27]Teitelbaum, J. T.. Values of $p$-adic $L$-functions and a $p$-adic Poisson kernel. Invent. Math. 101(2) (1990), 395–410.Google Scholar
[28]Tukia, P.. A rigidity theorem for Möbius groups. Invent. Math. 97(2) (1989), 405–431.Google Scholar
[29]Yue, C.. Mostow rigidity of rank $1$ discrete groups with ergodic Bowen–Margulis measure. Invent. Math. 125(1) (1996), 75–102.Google Scholar