Weil's local rigidity theorem

Remarks

A uniform lattice is finitely generated, so the topology on $\Hom(\Gamma, G)$ is that of a real algebraic set, given by the images of finitely many generators. Weil proved a general criterion for a homomorphism $\rho$ from a finitely generated group $\Gamma$ to $G$. Let $\Gamma$ act on the Lie algebra $\mathfrak{g}$ of $G$ through $\rho$ and the adjoint representation. If the first cohomology group $H^1(\Gamma, \mathfrak{g})$ vanishes, then $\rho$ is locally rigid. The tangent space to deformations of $\rho$ is the space of cocycles, and conjugations give the coboundaries, so vanishing of $H^1$ means that every infinitesimal deformation is a conjugation. He then proved this vanishing for uniform lattices under the hypotheses above. The hypothesis on $\mathrm{SL}_2(\R)$ is necessary. The fundamental group of a closed hyperbolic surface sits in $\mathrm{PSL}_2(\R)$ as a uniform lattice, and it has a continuous family of non-conjugate deformations, described by Teichmüller space. Garland and Raghunathan extended the result to non-uniform lattices in groups of real rank one other than those locally isomorphic to $\mathrm{SL}_2(\R)$ and $\mathrm{SL}_2(\C)$.

Going deeper

For a student

Local rigidity is the first of a chain of rigidity results for lattices. Mostow rigidity upgrades it to a global statement: an abstract isomorphism between lattices extends to the ambient groups. Margulis superrigidity goes further for higher rank, by controlling all homomorphisms of a lattice into other groups.