Realisation of groups as automorphism groups in permutational categories

Abstract

It is shown that in various categories, including many consisting of maps or hypermaps, oriented or unoriented, of a given hyperbolic type, or of coverings of a suitable topological space, 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.

Path partial groups

“…For example, while Graphs, the category of simple undirected graphs, is known to be universal [17,14,20] and this has been used to show that algebras over any field are finitely universal [10], deciding whether Galois extensions over the field of rational numbers is finitely universal, the Inverse Problem of Galois Theory [22], is still an open question. A compilation of some relevant achievements in identifying universal categories can be found in [19,Introduction].…”

Regular evolution algebras are universally finite

“…the category of differential graded algebras) is considered, mathematicians have been addressing the identification of finitely universal categories for almost a century. The reader is encouraged to consult [15,Introduction] for a (non exhaustive) account of milestones in this question.…”

Packable hyperbolic surfaces with symmetries

“…However, a packable surface does not need to have any symmetries at all. One should compare the following theorem with the results of [9,10].…”