Orderability of homology spheres obtained by Dehn filling
Gao, Xinghua
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https://hdl.handle.net/2142/105627
Description
Title
Orderability of homology spheres obtained by Dehn filling
Author(s)
Gao, Xinghua
Issue Date
2019-07-03
Director of Research (if dissertation) or Advisor (if thesis)
Dunfield, Nathan M.
Doctoral Committee Chair(s)
Leininger, Christopher J.
Committee Member(s)
Allen, Patrick B.
Bradlow, Steven B.
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(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.
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