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Abstract Let $$M$$ be a compact 3-manifold and $$\Gamma =\pi _1(M)$$. Work by Thurston and Culler–Shalen established the $${\operatorname{\textrm{SL}}}_2({\mathbb{C}})$$ character variety $$X(\Gamma )$$ as fundamental tool in the study of the geometry and topology of $$M$$. This is particularly the case when $$M$$ is the exterior of a hyperbolic knot $$K$$ in $S^3$. The main goals of this paper are to bring to bear tools from algebraic and arithmetic geometry to understand algebraic and number theoretic properties of the so-called canonical component of $$X(\Gamma )$$, as well as distinguished points on the canonical component, when $$\Gamma $$ is a knot group. In particular, we study how the theory of quaternion Azumaya algebras can be used to obtain algebraic and arithmetic information about Dehn surgeries, and perhaps of most interest, to construct new knot invariants that lie in the Brauer groups of curves over number fields.
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Coarse and fine geometry of the thurston metric
We study the geometry of the Thurston metric on the Teichmüller space of hyperbolic structures on a surface $$S$$ . Some of our results on the coarse geometry of this metric apply to arbitrary surfaces $$S$$ of finite type; however, we focus particular attention on the case where the surface is a once-punctured torus. In that case, our results provide a detailed picture of the infinitesimal, local, and global behavior of the geodesics of the Thurston metric, as well as an analogue of Royden’s theorem.
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- PAR ID:
- 10169325
- Publisher / Repository:
- Cambridge University Press
- Date Published:
- Journal Name:
- Forum of Mathematics, Sigma
- Volume:
- 8
- ISSN:
- 2050-5094
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/fms.2020.3
