Belyi functions of dessins d'enfants of genus 2 with 4 edges

Adrianov, N. M.; Shabat, G. B.

doi:10.1070/rm2005v060n06abeh004291

Cited by 8 publications

(24 citation statements)

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“…In order to find them we look for points in P that are moved an admissible distance by (at least) all the side-pairings generating K . Discerning the true face centers among these admissible points requires further arguments, mainly on the automorphisms of S. (1,5) in the particular case ( p, q, r ) = (5,5,5) The search of points mentioned above is computer aided, and the results will be presented through graphics produced with Mathematica [14] and the package [10]. Most times the results are visualized very clearly in the figures.…”

Section: Theorem 1 Two Unicellular Uniform Dessins Of the Same Type Imentioning

confidence: 99%

“…To retrieve a dessin from a Belyȋ function β (with normalized ramification values 0, 1 and ∞) one simply takes β −1 ([0, 1]). The white vertices can be taken to be β −1 (0) and the black vertices to be β −1 (1). As for β −1 (∞), it consists of one point inside each face, called the face center.…”

Section: Introductionmentioning

confidence: 99%

“…6). The only points of R 1 which are admissible for γ (1,6) and γ (2,4) simultaneously are the origin, p 1 , and the same point x in the study of D 4 , which is translated less than d 1 by γ (3,10) . Note that by getting rid of z 1 we discard z 5 as well since they correspond to the same point on the surface.…”

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confidence: 99%

“…5 where the two components of C d 1 (γ (2,4) ) are displayed), so by Lemma 3 there can be no admissible point in the convex sub-polygon generated by such points and the origin (the shaded region in the figure). The only points in the remaining region which are translated an admissible distance by γ (2,4) and by γ (1,6) simultaneously are the origin, p 1 and some point x in the real axis. But the latter one is translated by γ (3,5) a non-admissible distance d 1 < ρ(x, γ (3,5) …”

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confidence: 99%

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Genus 2 Belyĭ surfaces with a unicellular uniform dessin

Girondo

Torres-Teigell

2011

Geom Dedicata

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A bicoloured graph embedded in a compact oriented surface and dividing it into a union of simply connected components (faces) is known as a dessin d'enfant. It is well known that such a graph determines a complex structure on the underlying topological surface, but a given compact Riemann surface may correspond to different dessins. In this paper we deal with all unicellular (one-faced) uniform dessins of genus 2 and their underlying Riemann surfaces. A dessin is called uniform if white vertices, black vertices and faces have constant degree, say p, q and r respectively. A uniform dessin d'enfant of type ( p, q, r ) on a given surface S corresponds to the inclusion of the torsion-free Fuchsian group K uniformizing S inside a triangle group ( p, q, r ). Hence the existence of different uniform dessins on S is related to the possible inclusion of K in different triangle groups. The main result of the paper states that two unicellular uniform dessins belonging to the same genus 2 surface must necessarily be isomorphic or obtained by renormalisation. The problem is approached through the study of the face-centers of the dessins. The displacement of such a point by the elements of K must belong to a prescribed discrete set of (hyperbolic) distances determined by the signature ( p, q, r ). Therefore looking for face-centers amounts to finding points correctly displaced by every element of K .

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Section: Theorem 1 Two Unicellular Uniform Dessins Of the Same Type Imentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Genus 2 Belyĭ surfaces with a unicellular uniform dessin

Girondo

Torres-Teigell

2011

Geom Dedicata

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“…Топологией своего графа и его вложения в ориентированную поверхность пара Бе-лого определяется однозначно с точностью до изоморфизма [1]. Далее 2-вычетом мероморфного квадратичного дифференциала Ω в точке P называется коэффициент…”

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