Ueber eine einfache Gruppe von 504 Operationen

Fricke, R.

doi:10.1007/bf01476163

Cited by 24 publications

(26 citation statements)

References 1 publication

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“…In the first one [4], by Burnside, it was shown that the simple group PSL 2 (8) has a (3, 2, 7)-presentation; according to a tradition of that era, no motivation for this result was given. In the second paper [18], Fricke constructed a Riemann surface of genus 7 with the automorphism group PSL 2 (8) of order 504 = 84 • 6, that is, of maximal size for that genus. Later on this surface was rediscovered by Macbeath [29].…”

Section: The Fricke-macbeath Curvementioning

confidence: 99%

Hurwitz groups as monodromy groups of dessins: several examples

Jones¹,

Zvonkin²

2020

Preprint

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We present a number of examples to illustrate the use of small quotient dessins as substitutes for their often much larger and more complicated Galois (minimal regular) covers. In doing so we employ several useful group-theoretic techniques, such as the Frobenius character formula for counting triples in a finite group, pointing out some common traps and misconceptions associated with them. Although our examples are all chosen from Hurwitz curves and groups, they are relevant to dessins of any type.

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Section: The Fricke-macbeath Curvementioning

confidence: 99%

Hurwitz groups as monodromy groups of dessins: several examples

Jones¹,

Zvonkin²

2020

Preprint

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“…This algorithm is used here together with an exact determination of the shortest lattice vector. As an example we consider the Fricke-Macbeath curve [16,31], a curve of genus 7 with the maximal number of automorphisms.…”

Section: Approximation To the Siegel Fundamental Domainmentioning

confidence: 99%

“…As an example we want to study the Riemann matrix of the Fricke-Macbeath surface [16,31], a surface of genus g = 7 with the maximal number 84(g − 1) = 504 of automorphisms. It can be defined via the algebraic curve (25) f (x, y) := 1 + 7yx + 21y 2 x 2 + 35x 3 y 3 + 28x 4 y 4 + 2x 7 + 2y 7 = 0.…”

Section: Examplementioning

confidence: 99%

Efficient computation of multidimensional theta functions

Frauendiener

Jaber

Klein

2019

Journal of Geometry and Physics

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An important step in the efficient computation of multi-dimensional theta functions is the construction of appropriate symplectic transformations for a given Riemann matrix assuring a rapid convergence of the theta series. An algorithm is presented to approximately map the Riemann matrix to the Siegel fundamental domain. The shortest vector of the lattice generated by the Riemann matrix is identified exactly, and the algorithm ensures that its length is larger than √ 3/2. The approach is based on a previous algorithm by Deconinck et al. using the LLL algorithm for lattice reductions. Here, the LLL algorithm is replaced by exact Minkowski reductions for small genus and an exact identification of the shortest lattice vector for larger values of the genus.

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“…Let S be the Riemann surface of genus 7 admitting 504 conformal automorphisms. This surface is known as the Fricke-Macbeath surface; see [8,14]. It is known that S underlies a regular map M of type {3, 7}, which is called the Fricke-Macbeath map.…”

Section: Patterns and Mirror Automorphismsmentioning

confidence: 99%

Mirrors of reflections of regular maps

Melekoğlu

2018

Ars Math. Contemp.

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A regular map M is an embedding of a finite connected graph into a compact surface S such that its automorphism group Aut + (M) acts transitively on the directed edges. A reflection of M fixes a number of simple closed geodesics on S, which are called mirrors. In this paper, we prove two theorems which enable us to calculate the total number of mirrors fixed by the reflections of a regular map and the lengths of these mirrors. Furthermore, by applying these theorems to Hurwitz maps, we obtain some interesting results. In particular, we find an upper bound for the number of mirrors on Hurwitz surfaces.

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Ueber eine einfache Gruppe von 504 Operationen

Cited by 24 publications

References 1 publication

Hurwitz groups as monodromy groups of dessins: several examples

Hurwitz groups as monodromy groups of dessins: several examples

Efficient computation of multidimensional theta functions

Mirrors of reflections of regular maps

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