#### Vol. 284, No. 2, 2016

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Spherical CR Dehn surgeries

### Miguel Acosta

Vol. 284 (2016), No. 2, 257–282
##### Abstract

Consider a three-dimensional cusped spherical $CR$ manifold $M$ and suppose that the holonomy representation of ${\pi }_{1}\left(M\right)$ can be deformed in such a way that the peripheral holonomy is generated by a nonparabolic element. We prove that, in this case, there is a spherical $CR$ structure on some Dehn surgeries of $M$. The result is very similar to R. Schwartz’s spherical $CR$ Dehn surgery theorem, but has weaker hypotheses and does not give the uniformizability of the structure. We apply our theorem in the case of the Deraux–Falbel structure on the figure eight knot complement and obtain spherical $CR$ structures on all Dehn surgeries of slope $-3+r$, for $r\in {ℚ}^{+}$ small enough.

##### Keywords
spherical CR, Dehn surgery, $(G,X)$-structures, figure-eight knot
##### Mathematical Subject Classification 2010
Primary: 32V05, 57M25, 57M50