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2017 On the universal deformations for ${\rm SL}_2$-representations of knot groups
Masanori Morishita, Yu Takakura, Yuji Terashima, Jun Ueki
Tohoku Math. J. (2) 69(1): 67-84 (2017). DOI: 10.2748/tmj/1493172129


Based on the analogies between knot theory and number theory, we study a deformation theory for ${\rm SL}_2$-representations of knot groups, following after Mazur's deformation theory of Galois representations. Firstly, by employing the pseudo-${\rm SL}_2$-representations, we prove the existence of the universal deformation of a given ${\rm SL}_2$-representation of a finitely generated group $\Pi$ over a perfect field $k$ whose characteristic is not 2. We then show its connection with the character scheme for ${\rm SL}_2$-representations of $\Pi$ when $k$ is an algebraically closed field. We investigate examples concerning Riley representations of 2-bridge knot groups and give explicit forms of the universal deformations. Finally we discuss the universal deformation of the holonomy representation of a hyperbolic knot group in connection with Thurston's theory on deformations of hyperbolic structures.


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Masanori Morishita. Yu Takakura. Yuji Terashima. Jun Ueki. "On the universal deformations for ${\rm SL}_2$-representations of knot groups." Tohoku Math. J. (2) 69 (1) 67 - 84, 2017.


Published: 2017
First available in Project Euclid: 26 April 2017

zbMATH: 06726842
MathSciNet: MR3640015
Digital Object Identifier: 10.2748/tmj/1493172129

Primary: 57M25
Secondary: 14D15 , 14D20

Keywords: Arithmetic topology , Character scheme , Deformation of a representation , knot group

Rights: Copyright © 2017 Tohoku University


Vol.69 • No. 1 • 2017
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