Genus two extremal surfaces: Extremal discs, isometries and Weierstrass points

A compact Riemann surface of genus g > 1 has different uniform dessins d'enfants of the same type if and only if its surface group S is contained in different conjugate Fuchsian triangle groups Δ and αΔα −1 . Tools and results in the study of these conjugates depend on whether Δ is an arithmetic triangle group or not. In the case when Δ is not arithmetic the possible conjugators are rare and easy to classify. In the arithmetic case, i.e. for Shimura curves, the problem is much more complicated, but the arithmetic of quaternion algebras controls the growth of the number of uniform dessins of given type with respect to the genus. This number grows at most as O(g 1/3 ) and this bound is sharp. As a tool, localization of the quaternion algebras and the representation of p-adic maximal orders as vertices of Serre-Bruhat-Tits trees turn out to be crucial. In low genera, the results shed a surprising new light on the uniformization of some classical curves like Klein's quartic and other Macbeath-Hurwitz curves.

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“…There is obviously as well a uniform dessin of type (2,4,7), as already noticed in [13]. The third example given in Sect.…”

Section: A Geometrical Descriptionmentioning

confidence: 49%

“…The remaining groups Δ (2,3,14) = N (Δ (3,3,7)) and Δ (2,7,4) = N (Δ(2, 7, 7)) can be arithmetically constructed as the extensions of Δ 0 (π) and Δ (7,7,7) by the Fricke involution.…”

Section: Commensurabilitymentioning

confidence: 99%

Shimura curves with many uniform dessins

2011

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“…The above result still holds for g ¼ 3 but proving that Y g has only one disc requires the use of techniques similar to those used in [2] or [7].…”

Section: Notementioning

confidence: 83%

“…The proof of this statement is too long to be included here and can be found in [7]. We content ourselves with presenting an Idea of the proof: There are two key ingredients.…”

Section: Uniqueness Of Extremal Discs When G !mentioning

confidence: 99%

On extremal Riemann surfaces and their uniformizing fuchsian groups

Girondo

González-Diez

2002

Glasgow Math. J.

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Abstract. Compact hyperbolic surfaces of given genus g containing discs of the maximum radius have been studied from various points of view. In this paper we connect these different approaches and observe some properties of the Fuchsian groups uniformizing both compact and punctured extremal surfaces. We also show that extremal surfaces of genera g ¼ 2; 3 may contain one or several extremal discs, while an extremal disc is necessarily unique for g ! 4. Along the way we also construct explicit families of extremal surfaces, one of which turns out to be free of automorphisms.2000 Mathematics Subject Classification. 30F35, 30F10.1. Introduction. Extremal surfaces, that is hyperbolic surfaces containing discs of maximum radius in given genus, appear in the literature from different points of view: as cycloidal groups (Petterson and Millington, see [13]), as central curves (Macbeath [12]), f rom the point of view of generic polygon side pairings (Fricke and Klein [4] or Jorgensen and Na¨a¨ta¨nen [11]) and finally as genuine extremal surfaces (Bavard [2]). In this paper we study some properties of extremal surfaces and the groups which uniformize them. Its content is as follows.In Section 2 we gather together the above mentioned points of view and show that they are all equivalent. The key characterization is that extremal surfaces of genus g arise, exactly, as semiregular covers of the sphere with three branch points of order ð2; 3; 12g À 6Þ. We also construct two explicit families X g , Y g of such surfaces.In Section 3 we show that the groups uniformizing extremal surfaces (resp. extremal surfaces with the center deleted, called cycloidal) are never normal in the triangle group of type ð2; 3; 12g À 6Þ (resp. in PSL 2 ðZÞ).In Sections 4, 5 we show that while extremal surfaces of genera g ¼ 2; 3 may contain several extremal discs, for g ! 4 extremal surfaces contain exactly one. In the case of genus 2 we, in fact, consider the eight extremal surfaces of genus 2, whose description goes back to Fricke and Klein ([4], see also [11]), and detect the number of centers each of them has. The uniqueness of discs when g ! 4 is obtained by linking extremality to arithmeticity (of the uniformizing groups). This result was presented in the note [6] but we include it here for the sake of completeness.

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“…Our problem is to show how many extremal discs are embedded in an extremal surface. In the case that S is orientable, it was solved for every genus g > 3 ( [2]), g = 2 ( [3] and [6]), and g = 3 ( [7]). In the case that S is non-orientable, we know the following for g > 6 and g = 3 ( [4]): an extremal surface of genus g > 6 contains a unique extremal disc; there exist 11 extremal surfaces of genus 3, and each of them contains at most two extremal discs.…”

Section: Introductionmentioning

confidence: 99%

Compact non-orientable surfaces of genus 4 with extremal metric discs

Nakamura

2009

Conform. Geom. Dyn.

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Abstract. A compact hyperbolic surface of genus g is said to be extremal if it admits an extremal disc, a disc of the largest radius determined by g. We know how many extremal discs are embedded in a non-orientable extremal surface of genus g = 3 or g > 6. We show in the present paper that there exist 144 non-orientable extremal surfaces of genus 4, and find the locations of all extremal discs in those surfaces. As a result, each surface contains at most two extremal discs. Our methods used here are similar to those in the case of g = 3.

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Genus two extremal surfaces: Extremal discs, isometries and Weierstrass points

Cited by 20 publications

References 10 publications

Shimura curves with many uniform dessins

Shimura curves with many uniform dessins

On extremal Riemann surfaces and their uniformizing fuchsian groups

Compact non-orientable surfaces of genus 4 with extremal metric discs

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