Berkovich Curves and Schottky Uniformization II: Analytic Uniformization of Mumford Curves

2020 ◽  
pp. 225-279
Author(s):  
Jérôme Poineau ◽  
Daniele Turchetti
Keyword(s):  
Schottky Uniformization

scholarly journals Schottky uniformizations of closed Riemann surfaces with Abelian groups of conformal automorphisms

1994 ◽  
Vol 36 (1) ◽  
pp. 17-32 ◽  
Author(s):  
Rubén A. Hidalgo
Keyword(s):  
Abelian Group ◽  
Riemann Surface ◽  
Riemann Surfaces ◽  
Abelian Groups ◽  
Its Region ◽  
Schottky Group ◽  
Schottky Groups ◽  
Schottky Uniformization ◽  
Closed Riemann Surface ◽  
Holomorphic Covering

Let us consider a pair (S, H) consisting of a closed Riemann surface S and an Abelian group H of conformal automorphisms of S. We are interested in finding uniformizations of S, via Schottky groups, which reflect the action of the group H. A Schottky uniformization of a closed Riemann surface S is a triple (Ώ, G, π:Ώ→S) where G is a Schottky group with Ώ as its region ofdiscontinuity and π:Ώ→S is a holomorphic covering with G ascovering group. We look for a Schottky uniformization (Ώ, G, π:Ώ→S) of S such that for each transformation h in H there exists an automorphisms t of Ώ satisfying h ∘ π = π ∘ t.

scholarly journals SCHOTTKY UNIFORMIZATIONS OF SYMMETRIES

2013 ◽  
Vol 55 (3) ◽  
pp. 591-613 ◽  
Author(s):  
G. GROMADZKI ◽  
R. A. HIDALGO
Keyword(s):  
Riemann Surface ◽  
Kleinian Group ◽  
Riemann Surfaces ◽  
Algebraic Curve ◽  
Schottky Group ◽  
Structural Description ◽  
Schottky Groups ◽  
Real Algebraic Curve ◽  
Schottky Uniformization ◽  
Closed Riemann Surface

AbstractA real algebraic curve of genus g is a pair (S,〈 τ 〉), where S is a closed Riemann surface of genus g and τ: S → S is a symmetry, that is, an anti-conformal involution. A Schottky uniformization of (S,〈 τ 〉) is a tuple (Ω,Γ,P:Ω → S), where Γ is a Schottky group with region of discontinuity Ω and P:Ω → S is a regular holomorphic cover map with Γ as its deck group, so that there exists an extended Möbius transformation $\widehat{\tau}$ keeping Ω invariant with P o $\widehat{\tau}$=τ o P. The extended Kleinian group K=〈 Γ, $\widehat{\tau}$〉 is called an extended Schottky groups of rank g. The interest on Schottky uniformizations rely on the fact that they provide the lowest uniformizations of closed Riemann surfaces. In this paper we obtain a structural picture of extended Schottky groups in terms of Klein–Maskit's combination theorems and some basic extended Schottky groups. We also provide some insight of the structural picture in terms of the group of automorphisms of S which are reflected by the Schottky uniformization. As a consequence of our structural description of extended Schottky groups, we get alternative proofs to results due to Kalliongis and McCullough (J. Kalliongis and D. McCullough, Orientation-reversing involutions on handlebodies, Trans. Math. Soc. 348(5) (1996), 1739–1755) on orientation-reversing involutions on handlebodies.

scholarly journals Measure-theoretic rigidity for Mumford curves

2012 ◽  
Vol 33 (3) ◽  
pp. 851-869
Author(s):  
GUNTHER CORNELISSEN ◽  
JANNE KOOL
Keyword(s):  
Absolute Continuity ◽  
Riemann Surfaces ◽  
Canonical Embedding ◽  
Property A ◽  
Theoretic Property ◽  
Absolutely Continuous ◽  
Boundary Map ◽  
Schottky Uniformization ◽  
Special Fibers ◽  
Hyperbolic Riemann Surfaces

AbstractOne 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.

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