Local Rigidity of group actions: Past, Present, Future

TL;DR

This survey reviews the historical development, recent advances, and open problems in the study of local rigidity of group actions, highlighting key results and future research directions.

Contribution

It provides a comprehensive overview of the field, including detailed discussions of recent results, proofs, and a collection of conjectures and open questions.

Findings

Detailed discussion of recent results in local rigidity

Outlines of some proofs of key theorems

Compilation of conjectures and open problems

Abstract

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. {\em To Anatole Katok on the occasion of his 60th birthday.}