Duke Mathematical Journal

Local rigidity of hyperbolic 3-manifolds after Dehn surgery

Kevin P. Scannell

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It is well known that some lattices in ${\rm SO}(n,1)$ can be nontrivially deformed when included in ${\rm SO}(n+1,1)$ (e.g., via bending on a totally geodesic hypersurface); this contrasts with the (super) rigidity of higher rank lattices. M. Kapovich recently gave the first examples of lattices in ${\rm SO}(3,1)$ which are locally rigid in ${\rm SO}(4,1)$ by considering closed hyperbolic $3$-manifolds obtained by Dehn filling on hyperbolic two-bridge knots. We generalize this result to Dehn filling on a more general class of one-cusped finite volume hyperbolic $3$-manifolds, allowing us to produce the first examples of closed hyperbolic $3$-manifolds which contain embedded quasi-Fuchsian surfaces but are locally rigid in ${\rm SO}(4,1)$.

Article information

Duke Math. J., Volume 114, Number 1 (2002), 1-14.

First available in Project Euclid: 18 June 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M50: Geometric structures on low-dimensional manifolds
Secondary: 22E40: Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx] 57N16: Geometric structures on manifolds [See also 57M50]


Scannell, Kevin P. Local rigidity of hyperbolic 3-manifolds after Dehn surgery. Duke Math. J. 114 (2002), no. 1, 1--14. doi:10.1215/S0012-7094-02-11411-2. https://projecteuclid.org/euclid.dmj/1087575354

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