Generalized operations on hypermaps

Abstract

The algebraic theory of maps and hypermaps is summarized in Chapter 1.

There is a group of six invertible topological operations on maps which are induced by automorphisms of a certain Coxeter group that can be identified with an extended Fuchsian triangle group.  In Chapter 2 we study how these operations behave with respect to the property of orientability of maps, and we determine the orbits under the group of operations on reflexible torus maps of Euclidean type.

The corresponding groups of operations on hypermaps are infinite, and they partition the sets of symmetrical hypermaps having the same automorphism group into orbits.  In Chapter 3, given a group from one of several infinite families (cyclic, dihedral, affine general linear), we use the theory of T-systems to examine how the size of the orbits increase with the size of the group.

In Chapters 4 and 5 we generalize the concept of (hyper)map operations by considering functors induced by more general homomorphisms between triangle groups and extended triangle groups.

The concept of a 2-dimensional algebraic map can be generalized to n dimensions, and the group of operations on n-dimensional maps has order 8 for n > 2.  In Chapter 6 we give a combinatorial description of these operations, and examine the orbits of small 3-maps and of certain reflexible n-torus maps.  We then consider a further generalization of operations as isomorphism-induced equivalence between categories of different map-like objects.  In particular, we exhibit a representation of orientable 3-maps without boundary by (unrestricted) hypermaps, and another of general 3-maps by a certain family of maps.

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