A primary concern of pure mathematicians is to develop abstractions of basic structures, often structures encountered in nature, so that we can undertake investigations that simultaneously illuminate a number of different aspects of things. One very successful example of these investigations is group theory, which is an abstraction of the basic notions of counting and symmetry. However, group theory is vast, and so we are forced to limit our field of view. Within group theory, a relatively recent and successful addition has been the idea of treating a group, the basic construct in group theory, as a geometric object, so that we can apply results and ideas both from abstract algebra and from geometry to understand the behavior of certain classes of groups. As a further refinement, we often will focus our attention on certain specific classes of groups whose behavior is both illuminating and provides a testing ground for ideas that can then be applied to larger and less-well-behaved classes of groups.
The mapping class groups of surfaces are a family of groups of symmetries of 2-dimensional objects, and so they offer a class of groups that have many advantages: they are concrete, in that we can draw pictures (at least in simple cases) of how the groups act; they are well-behaved with respect to the natural ways of measuring length, distance and angle on surfaces, and so they offer a forum for understanding the interaction of the algebra and geometry; and their behavior is not too straightforward. The behavior of mapping class groups naturally straddles the behavior of other well-behaved and well-studied classes of groups, in particular the linear groups, and as such the study of mapping class groups is naturally enriched by, and naturally enriches, the study of linear groups. This is one of the reasons that mapping class groups are, and have been, the subject of a broad range of mathematicians.
In this proposal, we examine a particular aspect of the behavior of mapping class groups, which is the determination of their finite subgroups. This is an area where the bevavior of the mapping class groups is wildly divergent from the behavior of linear groups. In particular, we wish to count the finite subgroups of mapping class groups. It has been known for many years that (up to a natural notion of sameness) mapping class groups have only finitely many finite subgroups, but there are no good estimates for even roughly how many there are. We recognize immediately that we will not be able to provide an exact count, because this is beyond current mathematical technology. However, we have an idea, which has not yet been fully exploited, that we will use to provide good estimates of how many finite subgroups there are in mapping class groups. The benefits of the work are to provide the global community of researchers specific information about the behavior of the finite subgroups of mapping class groups, which will then increase our understanding of how these groups behave, which will then enrich the study of these and similar groups.
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