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Dunfield, Nathan M. Leininger, Christopher J. Allen, Patrick B. Bradlow, Steven B. Gao, Xinghua 2019-11-26T20:33:46Z 2019-11-26T20:33:46Z 2019-07-03 2019-08 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. Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2019-11-26 without embargo terms The student, Xinghua Gao, accepted the attached license on 2019-06-30 at 00:17. The student, Xinghua Gao, submitted this Dissertation for approval on 2019-06-30 at 00:54. This Dissertation was approved for publication on 2019-07-03 at 10:46. DSpace SAF Submission Ingestion Package generated from Vireo submission #14111 on 2019-11-26 at 12:50:18 Made available in DSpace on 2019-11-26T20:33:46Z (GMT). No. of bitstreams: 2 GAO-DISSERTATION-2019.pdf: 1264775 bytes, checksum: 83c0ef9199136d2a4c99ecfed00cdafe (MD5) LICENSE.txt: 4208 bytes, checksum: 0dba3989f7d21b66e16c3dcb1988bf38 (MD5) Previous issue date: 2019-07-03 application/pdf http://hdl.handle.net/2142/105627 en Copyright 2019 Xinghua Gao left-orderable group $\widetilde{PSL_2\mathbb{R}}$ representation Orderability of homology spheres obtained by Dehn filling text text University of Illinois at Urbana-Champaign Mathematics Mathematics Dissertation Ph.D.