The genera, reflexibility and simplicity of regular maps

Conder, Marston; Širáň‬, Jozef; Tucker, Thomas W.

doi:10.4171/jems/200

Cited by 43 publications

(84 citation statements)

References 20 publications

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“…It would be interesting to characterise the genera with this property. Conder, Siráň and Tucker [5] have shown that there are no chiral maps of genus g = p+1 for primes p such that p − 1 is not divisible by 3, 5 or 8, but finding a similar result for hypermaps would seem to be more difficult.…”

Section: Casementioning

confidence: 99%

Chiral covers of hypermaps

Jones

2015

Ars Math. Contemp.

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Section: Casementioning

confidence: 99%

Chiral covers of hypermaps

Jones

2015

Ars Math. Contemp.

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“…The corresponding quasiplatonic surfaces X, of genus p+1, are constructed in §3, Examples (i) -(iii) [1], and the associated maps, regarded by duality as dessins of type q, 2, r , are studied in Theorem 3.1 of [5].…”

Section: Problems Conjectures and Infinite Familiesmentioning

confidence: 99%

Galois actions on regular dessins of small genera

Conder¹,

Jones²,

Streit³

et al. 2013

Rev. Mat. Iberoam.

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Abstract. Dessins d'enfants can be regarded as bipartite graphs embedded in compact orientable surfaces. According to Grothendieck and others, a dessin uniquely determines a complex structure on the surface, and even an algebraic structure (as a projective algebraic curve defined over a number field). The general problem of how to determine all properties of the curve from the combinatorics of the dessin is far from being solved. For regular dessins, which are those having an edge-transitive automorphism group, the situation is easier: currently available methods in combinatorial and computational group theory allow the determination of the fields of definition for all curves with regular dessins of genus 2 to 18. The many facets of dessinsOne may introduce dessins d'enfants as hypermaps on compact oriented 2-manifolds. These objects, first studied in genus 0 by Cori in [9], have several equivalent algebraic or topological definitions. Topologically they are a generalisation of maps, whose underlying graphs are replaced with hypergraphs, in which hyperedges are allowed to be incident with any finite number of hypervertices and hyperfaces. One way to visualise this concept is to represent a hypermap D by its Walsh map W (D) , a connected bipartite graph B embedded in a compact oriented surface, dividing it into simply connected cells [31]. In the language of hypermaps, the white and black vertices represent the hypervertices and hyperedges of D, the edges represent incidences between them, and the cells represent the hyperfaces.There are several other ways to define dessins. There is a group theoretic description of maps and hypermaps by their (hyper)cartographic groups; see, for example, [18], [19] or the introduction of [20]. The latter are the monodromy groups of the functions about to be introduced, and give a link to function theory on Riemann surfaces and algebraic geometry on curves. Playing a key role are

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“…In [2], for example, some ground-breaking work by Breda, Nedela, andŠiráň [2] showed that there are no regular maps at all on non-orientable surfaces of Euler characteristic −p where p > 13 is prime and p ≡ 1 mod 12. We took a different approach in [13], to obtain a shorter proof of the latter, and also to prove that:…”

Section: Remaining Gaps In the Spectrummentioning

confidence: 99%

The symmetric genus spectrum of finite groups

Conder

Tucker

2011

Ars Math. Contemp.

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The symmetric genus of the finite group G, denoted by σ(G), is the smallest nonnegative integer g such that the group G acts faithfully on a closed orientable surface of genus g (not necessarily preserving orientation). This paper investigates the question of whether for every non-negative integer g, there exists some G with symmetric genus g. It is shown that that the spectrum (range of values) of σ includes every non-negative integer g ≡ 8 or 14 mod 18, and moreover, if a gap occurs at some g ≡ 8 or 14 modulo 18, then the prime-power factorization of g − 1 includes some factor p e ≡ 5 mod 6. In fact, evidence suggests that this spectrum has no gaps at all.

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The genera, reflexibility and simplicity of regular maps

Cited by 43 publications

References 20 publications

Chiral covers of hypermaps

Chiral covers of hypermaps

Galois actions on regular dessins of small genera

The symmetric genus spectrum of finite groups

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