Spherical CR Dehn Surgery

TL;DR

This paper proves that under certain deformations of the holonomy representation, some Dehn surgeries on a cusped spherical CR manifold admit spherical CR structures, extending previous results with weaker assumptions.

Contribution

It establishes a new spherical CR Dehn surgery theorem with weaker hypotheses and applies it to the Figure Eight knot complement.

Findings

Spherical CR structures exist on certain Dehn surgeries under deformed holonomy.

The theorem generalizes Schwartz's result by relaxing conditions.

Applied to the Figure Eight knot, structures are found on all slopes near -3.

Abstract

Consider a three dimensional cusped spherical $\mathrm{CR}$ manifold $M$ and suppose that the holonomy representation of $\pi_1(M)$ can be deformed in such a way that the peripheral holonomy is generated by a non-parabolic element. We prove that, in this case, there is a spherical $\mathrm{CR}$ structure on some Dehn surgeries of $M$. The result is very similar to R. Schwartz's spherical $\mathrm{CR}$ Dehn surgery theorem, but has weaker hypotheses and does not give the unifomizability of the structure. We apply our theorem in the case of the Deraux-Falbel structure on the Figure Eight knot complement and obtain spherical $\mathrm{CR}$ structures on all Dehn surgeries of slope $-3 + r$ for $r \in \mathbb{Q}^{+}$ small enough.