The Liftable Mapping Class Group
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Date
2017-07-19
Authors
Ghaswala, Tyrone
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Publisher
University of Waterloo
Abstract
Broadly, this thesis lies at the interface of mapping class groups and covering spaces. The foundations of this area were laid down in the early 1970s by Birman and Hilden. Building on these foundations, there has been a plethora of results, especially in the context of a particular family of branched covering spaces over the sphere. We call these covers the hyperelliptic covering spaces.
One of the reasons the hyperelliptic covers provide such fertile ground for research is that every homeomorphism of a marked sphere lifts to a homeomorphism of the covering space. Rephrasing this, the liftable mapping class group coincides with the entire mapping class group of a marked sphere. Since the mapping class group of a marked sphere is well understood, this understanding can be lifted to help understand a particular subgroup of the mapping class group of the covering space.
However, for a general covering space, the liftable mapping class group does not coincide with the mapping class group of the base space. Instead, it is a finite index subgroup. This thesis is devoted to studying the liftable mapping class group in contexts other than the hyperelliptic covers.
Chapter 2 provides the necessary preliminaries for the rest of the thesis. Chapter 3 classifies cyclic branched covers of the sphere with the property that the liftable mapping class group coincides with the mapping class group of the marked sphere. Chapter 4 studies the liftable mapping class group for a family of cyclic branched covers over the sphere, called balanced superelliptic covers. We find an explicit finite presentation for the liftable mapping class groups corresponding to the balanced superelliptic covers, compute the indexes of the liftable mapping class groups, and compute their abelianizations.
In Chapter 5 we study an infinite family of cyclic branched covers over a torus. The liftable mapping class groups corresponding to this family are all subgroups of the mapping class group of a twice marked torus. We prove that the intersection of any two of these liftable mapping class groups is also a liftable mapping class group, and the subgroup generated by any two is again a liftable mapping class group. In a few special cases we find a finite generating set, and for some of those, an explicit finite presentation.
Finally, in Chapter 6 we study the liftable mapping class group for covers of surfaces with boundary. Given a covering space of a surface with boundary, we characterize the corresponding liftable mapping class group by the action of its members on a particular fundamental groupoid of the base surface.
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Keywords
Mapping class groups