Left-orderability, Cyclic Branched Covers and Representations of the Knot Group
نویسنده
چکیده
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Chapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Left-orderable groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Left-orderability of 3-manifold groups . . . . . . . . . . . . . . . . . . 3 1.3 Summary of the Dissertation . . . . . . . . . . . . . . . . . . . . . . . 7 Chapter 2: Left-Orderability and Cyclic Branched Covers . . . . . . . . . . . . . . . . . . . 8 2.1 Knots in S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Cyclic branched covers of S . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 Proof of Lemma 2.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4 Left-orderability and cyclic branched covers . . . . . . . . . . . . . . 17 Chapter 3: Applications to Two-Bridge Knots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.1 SL(2,C)-representations of two-bridge knot groups . . . . . . . . . . 20 3.2 Proof of Theorem 3.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.3 Further discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Chapter 4: Application to Satellite Knots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.1 Lemmas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.2 Left-orderability and Cyclic branched covers over satellite knots . . . 32 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Appendix: Permission for Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
منابع مشابه
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We show that several torsion free 3-manifold groups are not left-orderable. Our examples are groups of cyclic branched coverings of S branched along links. The figure eight knot provides simple nontrivial examples. The groups arising in these examples are known as Fibonacci groups which we show not to be left-orderable. Many other examples of non-orderable groups are obtained by taking 3-fold b...
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