Rigidity of representations in SO(4,1) for Dehn fillings on 2-bridge knots

Abstract

We prove that, for a hyperbolic two-bridge knot, infinitely many Dehn fillings are rigid in SO 0 (4, 1). Here rigidity means that any discrete and faithful representation in SO 0 (4, 1) is conjugate to the holonomy representation in SO 0 (3, 1). We also show local rigidity for almost all Dehn fillings.

Conway spheres as ideal points of the character variety

“…Figure 1. A different choice of isometry a ∈ P SL(2, C) differs by an isometry that preserves the oriented line from 0 to ∞, hence it commutes with the isometry in (5). Note also that the inverse in SL(2, C) of the matrix in the above identity coincides with its opposite, which is in accordance with the chosen convention.…”

Volume rigidity ad ideal points of the character variety of hyperbolic 3-manifolds

“…Similarly, when m > k ≥ 3 the space of representations is rich. For instance one can bend along geodesic hypersurfaces (see [Apa90]), but also purely parabolic deformations are possible (see [FP08] for the study of deformations in H 4 of complements of hyperbolic two-bridge knots).…”

Reidemeister torsion, hyperbolic three-manifolds, and character varieties

“…When k 1 = k 2 , ̺ k,k may be not acyclic (for instance when it contains a totally geodesic surface [64]), but sometimes it can be acyclic (e.g. almost all Dehn fillings on two bridge knots for k = 1 by [45,88,28] or on the figure eight knot for k = 2 by [42]). When ̺ k,k is acyclic, then τ k,k (M 3 , σ) is also well defined.…”