On a canonical construction of tesselated surfaces via finite group theory, Part II

Abstract

This paper is the second part of a two-part study of an elementary functorial construction of tesselated surfaces from finite groups. This elementary construction was discussed in the first part and generally results in a large collection of tesselated surfaces per group, for example when the construction is applied to \Sigma_6 it yields 4477 tesselated surfaces of 27 distinct genus and even more varieties of tesselation cell structure. These tesselations are face and edge transitive and consist of closed cell structures. In this paper, we continue to study the distribution of these surfaces in various groups and some interesting resulting tesselations with the aid of computer computations. 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.