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On a canonical construction of a singular manifold from a finite group

In this paper we study an elementary functorial construction from the category of finite non-abelian groups to a category of singular compact, oriented 2-manifolds. After a desingularization process this construction results in a collection of compact, connected, oriented tesselated smooth surfaces equipped with a closed-cell structure which is face and edge transitive and which has at most 2 orbits of vertices. These tesselated surfaces can also be viewed as abstract 3-polytopes (or as graph embeddings in the corresponding surface) which are either equivar or dual to abstract quasiregular polytopes. This construction is elementary and bypasses the use of Fuchsian groups and furthermore generally results in a large collection of tesselated surfaces per group, for example when the construction is applied to Σ6 it yields 4477 tesselated surfaces of 27 distinct genus and even more varieties of tesselation cell structure. We study the distribution of these surfaces in various groups and some interesting resulting tesselations. We also show that extensions of groups result in branched coverings between the component surfaces in their decompositions. Finally we exploit functoriality to obtain interesting faithful, orientation preserving actions of subquotients of these groups and their automorphism groups on these surfaces and in the corresponding mapping class groups.