Tohoku Mathematical Journal

On the universal deformations for ${\rm SL}_2$-representations of knot groups

Masanori Morishita, Yu Takakura, Yuji Terashima, and Jun Ueki

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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.

Article information

Tohoku Math. J. (2), Volume 69, Number 1 (2017), 67-84.

First available in Project Euclid: 26 April 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}
Secondary: 14D15: Formal methods; deformations [See also 13D10, 14B07, 32Gxx] 14D20: Algebraic moduli problems, moduli of vector bundles {For analytic moduli problems, see 32G13}

Deformation of a representation Character scheme Knot group Arithmetic topology


Morishita, Masanori; Takakura, Yu; Terashima, Yuji; Ueki, Jun. On the universal deformations for ${\rm SL}_2$-representations of knot groups. Tohoku Math. J. (2) 69 (2017), no. 1, 67--84. doi:10.2748/tmj/1493172129.

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