Joining dessins together

Jones, Gareth

doi:10.48550/arxiv.1810.03960

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2018

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“…In terms of the corresponding maps D i , this is a connected sum operation, in which the two surfaces are joined across cuts between the free ends of the free edges representing the fixed points α i and β i ; in particular, if D i has genus g i then D 1 (1)D 2 has genus g 1 + g 2 . (Further details about this and more general joining operations on dessins will appear in [25]. )…”

Section: G Hmentioning

confidence: 99%

Realisation of groups as automorphism groups in categories

Jones

2018

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It is shown that in various categories, including many consisting of maps or hypermaps, oriented or unoriented, of a given hyperbolic type, every countable group A is isomorphic to the automorphism group of uncountably many non-isomorphic objects, infinitely many of them finite if A is finite. In particular, the latter applies to dessins d'enfants, regarded as finite oriented hypermaps. The proof, involving maximal subgroups of various triangle groups, yields a simple construction of a regular map whose automorphism group contains an isomorphic copy of every finite group.

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Section: G Hmentioning

confidence: 99%