Local rigidity of group actions: past, present, future

Summary

This survey aims to cover the motivation for and history of the study of local rigidity of group actions. There is a particularly detailed discussion of recent results, including outlines of some proofs. The article ends with a large number of conjectures and open questions and aims to point to interesting directions for future research.

Let Γ be a finitely generated group, D a topological group, and π : Γ→D a homomorphism. We wish to study the space of deformations or perturbations of π. Certain trivial perturbations are always possible as soon as D is not discrete, namely we can take d π d^-1 where d is a small element of D.

DEFINITION 1.1. Given a homomorphism π : Γ →D, we say is locally rigid if any other homomorphism π′ which is close to π is conjugate to by π a small element of D.

We topologize Hom(Γ;D) with the compact open topology which means that two homomorphisms are close if and only if they are close on a generating set for Γ . If D is path connected, then we can define deformation rigidity instead, meaning that any continuous path of representations π _t starting at π is conjugate to the trivial path π _t = π by a continuous path d_t in D with d₀ being the identity in D. If D is an algebraic group over ℝ or ℂ, it is possible to prove that deformation rigidity and local rigidity are equivalent since Hom(Γ,D) is an algebraic variety and the action of D by conjugation is algebraic; see [Mu], for example. For D infinite dimensional and path-connected, this equivalence is no longer clear.

The study of local rigidity of lattices in semisimple Lie groups is probably the beginning of the general study of rigidity in geometry and dynamics, a subject that is by now far too large for a single survey. See [Sp1] for the last attempt at a comprehensive survey and [Sp2] for a more narrowly focused updating of that survey. Here we abuse language slightly by saying a subgroup is locally rigid if the defining embedding is locally rigid as a homomorphism. See subsection 3.1 for a brief history of local rigidity of lattices and some discussion of subsequent developments that are of particular interest in the study of rigidity of group actions.