Realisation of groups as automorphism groups in categories

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 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.

A short proof of Greenberg's Theorem

“…5 This proof of Greenberg's Theorem is adapted from a proof in [10,Theorem 3(a)] that for any hyperbolic triple (l, m, n) there are ℵ 0 non-isomorphic dessins of type (l, m, n) with a given finite automorphism group A. (See also [9] for related results by Hidalgo on realising groups as automorphism groups of dessins.)…”

“…Again, the proof is rather delicate (but see [1] for a more elementary geometric proof). In [10] it is shown that for many hyperbolic triples (including all of non-cocompact type and many of cocompact type), every countable group can be realised as the automorphism group of 2 ℵ 0 non-isomorphic oriented hypermaps of that type. It would be interesting to try to deduce Greenberg's result for countable groups from this.…”

Telescopic groups and symmetries of combinatorial maps

“…Soon after a draft of this paper appeared on the arXiv, the authors were notified by Gareth A. Jones that his paper [12] contains similar results for a wider class of group. In particular, by [12,Theorem 3], all hyperbolic (extended) triangle groups are shown to be "finitely abundant", which is equivalent to being telescopic for the non-compact ones among them. Also, all subgroups produced in [12,Theorem 3] are, in fact, torsion-free.…”

“…In particular, by [12,Theorem 3], all hyperbolic (extended) triangle groups are shown to be "finitely abundant", which is equivalent to being telescopic for the non-compact ones among them. Also, all subgroups produced in [12,Theorem 3] are, in fact, torsion-free. The methods used in [12] and in our paper differ substantially, as well as the emphasis in our work is on the quantitative aspects, such as counting of combinatorial objects with given symmetries.…”

Prescribing Symmetries and Automorphisms for Polytopes

“…The question whether or not a given group is the automorphism group or symmetry group of a geometric, combinatorial, algebraic, or topological structure of a specified kind has been studied quite extensively. For a recent article describing the common characteristics of the approaches see the recent article [Jon18] by Jones.…”