Automorphism groups of maps, hypermaps and dessins

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.

Finite simple automorphism groups of edge-transitive maps

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.

Automorphism groups of superextensions of groups

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 we study isomorphisms of superextensions of groups and prove that two groups are isomorphic if and only if their superextensions are isomorphic. Also we describe the automorphism groups of superextensions of all groups of cardinality $leq 5$.

Residual properties of automorphism groups of (relatively) hyperbolic groups

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 automorphisms preserving the peripheral structure is residually finite. We also show that Out(G) is virtually p-residually finite for every prime p if G is one-ended and toral relatively hyperbolic, or infinitely-ended and virtually p-residually finite.

Finite Groups with Abelian Automorphism Groups: A Survey

In this article we present an extensive survey on the developments in the theory of non-abelian finite groups with abelian automorphism groups, and pose some problems and further research directions.