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A detailed proof is given of a theorem describing the centraliser of a transitive permutation group, with applications to automorphism groups of objects in various categories of maps, hypermaps, dessins, polytopes and covering spaces, where the automorphism group of an object is the centraliser of its monodromy group. An alternative form of the theorem, valid for finite objects, is discussed, with counterexamples based on Baumslag--Solitar groups to show how it fails more generally. The automorphism groups of objects with primitive monodromy groups are described, as are those of non-connected objects.
Building on earlier results for regular maps and for orientably regular chiral maps, we classify the non-abelian finite simple groups arising as automorphism groups of maps in each of the 14 Graver-Watkins classes of edge-transitive maps.
It is known that every finite group can be represented as the full group of automorphisms of a suitable compact dessin denfant. In this paper, we give a constructive and easy proof that the same holds for any countable group by considering non-compac
The superextension $lambda(X)$ of a set $X$ consists of all maximal linked families on $X$. Any associative binary operation $*: Xtimes X to X$ can be extended to an associative binary operation $*: lambda(X)timeslambda(X)tolambda(X)$. In the paper w
We show that Out(G) is residually finite if G is a one-ended group that is hyperbolic relative to virtually polycyclic subgroups. More generally, if G is one-ended and hyperbolic relative to proper residually finite subgroups, the group of outer auto