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 Title: Orderability of homology spheres obtained by Dehn filling Author(s): Gao, Xinghua Director of Research: Dunfield, Nathan M. Doctoral Committee Chair(s): Leininger, Christopher J. Doctoral Committee Member(s): Allen, Patrick B.; Bradlow, Steven B. Department / Program: Mathematics Discipline: Mathematics Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): left-orderable group $\widetilde{PSL_2\mathbb{R}}$ representation Abstract: In my thesis, I study left-orderability of $\mathbb{Q}$-homology spheres. I use $\widetilde{PSL_2\mathbb{R}}$ representations as a tool. First, I showed this tool has its limitations by constricting a series of $\mathbb{Z}$-homology spheres with potentially left-orderable fundamental groups but no non trivial $\widetilde{PSL_2\mathbb{R}}$ representations. However, this tool is still useful in most cases. With $\widetilde{PSL_2\mathbb{R}}$ representations, I construct the holonomy extension locus of a $\mathbb{Q}$-homology solid torus which is an analog of its translation extension locus. Using extension loci, I study $\mathbb{Q}$-homology 3-spheres coming from Dehn fillings of $\mathbb{Q}$-homology solid tori and construct intervals of orderable Dehn fillings. Issue Date: 2019-07-03 Type: Text URI: http://hdl.handle.net/2142/105627 Rights Information: Copyright 2019 Xinghua Gao Date Available in IDEALS: 2019-11-26 Date Deposited: 2019-08
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