Calabi–Weil rigidity

Builds on Weil's local rigidity theorem Lattice in a Lie group Group homomorphism

Statement

Let $n \ge 3$ and let $G = \mathrm{SO}_0(n, 1)$ be the group of orientation-preserving isometries of hyperbolic space $\mathbb{H}^n$ . Let $\Gamma \subset G$ be a discrete torsion-free subgroup such that the hyperbolic manifold $M = \Gamma \backslash \mathbb{H}^n$ is compact. Then the inclusion $\Gamma \to G$ is locally rigid. That is, let $\rho \colon \Gamma \to G$ be a homomorphism close enough to the inclusion on a finite generating set of $\Gamma$ . Then $\rho$ is conjugate to the inclusion by an element of $G$ . Equivalently, the hyperbolic structure of $M$ admits no nontrivial small deformations.

Remarks

Calabi proved this for compact hyperbolic manifolds, and Weil extended it to all uniform lattices in semisimple groups. It follows from Weil's local rigidity theorem, since $\Gamma$ is a uniform lattice in $G$ , which for $n \ge 3$ is simple, has trivial center and is not locally isomorphic to $\mathrm{SL}_2(\R)$ . The hypothesis $n \ge 3$ is necessary: for $n = 2$ the hyperbolic structures on a closed surface of genus $g \ge 2$ form a Teichmüller space of real dimension $6g - 6$ . For $n = 3$ compactness is also needed: hyperbolic structures on cusped $3$ -manifolds deform to incomplete structures, which give hyperbolic Dehn filling.

Going deeper

For a student

This theorem is the local version of Mostow rigidity, which states that the hyperbolic structure is determined by the fundamental group globally. In the language of representations, the point of $\Hom(\Gamma, G)/G$ defined by $M$ is isolated. The proof goes through the vanishing of the cohomology group $H^1(\Gamma, \mathfrak{g})$ with coefficients in the Lie algebra of $G$ .

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Tag	`X4DZR72B6N`
URI	`m2://X4DZR72B6N`
Kind	statement, corollary
Writing	W0 stub
MSC	53C24, 22E40, 53C35
Status	This copy of the site has no build data. Run `m2 status X4DZR72B6N`.
About	Lattice in a Lie group
Uses	Weil's local rigidity theorem, Lattice in a Lie group, Group homomorphism