Flexing closed hyperbolic manifolds. (Q2464446)

As remarked in [\textit{R. E. Schwartz}, Ann. Math. (2) 153, No. 3, 533--598 (2001; Zbl 1055.20040)], it is a basic problem to understand how discrete faithful representations \(\rho:\Gamma\to G_0\) can be deformed if one extends the Lie group \(G_0\) to a larger group \(G_1\). The best understood example of this is the case of quasi-Fuchsian deformation, where \(\Gamma\) is the fundamental group of a closed orientable surface and the Lie groups are \(G_0= \text{PSL}(2,\mathbb{R})\) and \(G_1= \text{PSL}(2,\mathbb{C})\). By considering the Klein model one sees that a hyperbolic structure is in particular a projective structure. The authors investigate the following question: Can the hyperbolic structure on a closed hyperbolic 3-manifold be deformed to a nontrivial real projective structure? Their first result is that with a mild smoothness hypothesis, the existence of such real projective deformations is in some sense equivalent to the existence of a deformation of the representation into the isometry group of complex hyperbolic space. More precisely, they prove the following result: assume that \(\rho: \pi_1(M)\to \text{SO}_0(n,1)\) is a representation of a closed hyperbolic \(n\)-manifold \(M\) which is a smooth point of the representation variety \(V= \Hom(\pi_1(M), \text{PGL}(n+ 1,\mathbb{R}))\). Then \(\rho\) is a smooth point of \(\Hom(\pi_1(M), PU(n, 1))\), and \[ \dim_{\mathbb{R}}(\Hom(\pi_1(M), PU(n, 1)))= \dim_{{\mathbb{R}}}\Hom(\pi_1(M), \text{PGL}(n+ 1,\mathbb{R})) \] near \(\rho\). Moreover, it turns out that if the original representation is the discrete faithful representation, then sufficiently close to \(\rho\) in \(\Hom(\pi_1(M), PU(n,1))\), the deformed representations are also discrete and faithful. Some examples complete this fine paper.