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Measure-theoretic rigidity for Mumford curves

Published online by Cambridge University Press:  17 April 2012

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)
Corresponding

Abstract

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.

Type
Research Article
Copyright
Copyright © 2012 Cambridge University Press

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