TY - JOUR
T1 - Measure-theoretic rigidity for Mumford curves
AU - Cornelissen, G.L.M.
AU - Kool, J.
PY - 2012
Y1 - 2012
N2 - 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.
AB - 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.
U2 - 10.1017/S0143385712000016
DO - 10.1017/S0143385712000016
M3 - Article
SN - 0143-3857
JO - Ergodic Theory and Dynamical Systems
JF - Ergodic Theory and Dynamical Systems
ER -