We explore the descriptive set theory of subgroups of fundamental groups, giving theorems regarding dichotomies on cardinality. In paticular, we show that the quotient of the fundamental group of a path connected, locally path connected Polish space by a normal subgroup which is sufficiently eay to describe topologically (of ``nice' pointclass having the property of Baire) is either countable or of cardinality continuum. In case the space is compact, countability of the quotient implies the quotient is finitely generated. We give upper bounds on the complexity of some subgroups, such as the shape kernel and the Spanier group. Applications to the normal generation of groups are given, as well as an application to covering space theory. We present an array of theorems regarding the first homology of Peano continua. We also demonstrate the existence of subgroups of all but finitely many additive and multiplicative Borel types. We prove that torsion-free word hyperbolic groups are n-slender, and the class of n-slender groups is closed under graph products.
| Identifer | oai:union.ndltd.org:VANDERBILT/oai:VANDERBILTETD:etd-03252016-151756 |
| Date | 09 April 2016 |
| Creators | Corson, Samuel Mark |
| Contributors | Mark V. Sapir, C. Bruce Hughes, Michael L. Mihalik, Denis V. Osin, Paul D. Sheldon |
| Publisher | VANDERBILT |
| Source Sets | Vanderbilt University Theses |
| Language | English |
| Detected Language | English |
| Type | text |
| Format | application/pdf |
| Source | http://etd.library.vanderbilt.edu/available/etd-03252016-151756/ |
| Rights | restricted, I hereby certify that, if appropriate, I have obtained and attached hereto a written permission statement from the owner(s) of each third party copyrighted matter to be included in my thesis, dissertation, or project report, allowing distribution as specified below. I certify that the version I submitted is the same as that approved by my advisory committee. I hereby grant to Vanderbilt University or its agents the non-exclusive license to archive and make accessible, under the conditions specified below, my thesis, dissertation, or project report in whole or in part in all forms of media, now or hereafter known. I retain all other ownership rights to the copyright of the thesis, dissertation or project report. I also retain the right to use in future works (such as articles or books) all or part of this thesis, dissertation, or project report. |
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