Realisation of groups as automorphism groups in categories
arXiv:1807.00547
Abstract
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 is isomorphic to the automorphism group of uncountably many non-isomorphic objects, infinitely many of them finite if is finite. In particular, this 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.
21 pages, 5 figures. In this version, definitions and theorems have been strengthened to realise each group as the automorphism group of uncountably many objects in a given category, instead of just one or infinitely many (as in versions 1 and 2). References to analogous results in the literature have been added, and a few typos have been corrected