Tohoku Mathematical Journal
2017
March
SECOND SERIES VOL. 69, NO. 1

Tohoku Math. J.
69 (2017), 67-84

Title ON THE UNIVERSAL DEFORMATIONS FOR ${\rm SL}_2$-REPRESENTATIONS OF KNOT GROUPS

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

(Received December 1, 2014)
Abstract. 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.

Mathematics Subject Classification. Primary 57M25; Secondary 14D15, 14D20.
Key words and phrases. Deformation of a representation, Character scheme, Knot group, Arithmetic topology.

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