Enumerating Regular Objects associated with Suzuki Groups

We use the M\"obius function of the simple Suzuki group Sz(q) to enumerate regular objects such as maps, hypermaps, dessins d'enfants and surface coverings with automorphism groups isomorphic to Sz(q).Comment: 20 page

Exotic behaviour of infinite hypermaps

This is a survey of infinite hypermaps, and of how they can be constructed by using examples and techniques from combinatorial group theory, with particular emphasis on phenomena which have no analogues for finite hypermaps.<br/

Injectivity radius of representations of triangle groups and planar width of regular hypermaps

We develop a rigorous algebraic background for representations of triangle groups in linear groups over algebras arising from factor rings of multivariate polynomial rings. This is then used to substantially improve the existing bounds on the order of epimorphic images of triangle groups with a given injectivity radius and, analogously, the size of the associated hypermaps of a given type with a given planar width

Generalised Fermat Hypermaps and Galois Orbits

We consider families of quasiplatonic Riemann surfaces characterised by the fact that -- as in the case of Fermat curves of exponent $n$ -- their underlying regular (Walsh) hypermap is the complete bipartite graph $K_{n,n}$, where $n$ is an odd prime power. We will show that all these surfaces, regarded as algebraic curves, are defined over abelian number fields. We will determine the orbits under the action of the absolute Galois group, their minimal fields of definition, and in some easier cases also their defining equations. The paper relies on group-- and graph--theoretic results by G. A. Jones, R. Nedela and M.\v{S}koviera about regular embeddings of the graphs $K_{n,n}$ [JN\v{S}] and generalises the analogous question for maps treated in [JStW], partly using different methods.Comment: 14 pages, new version with extended introduction, minor corrections and updated reference