Groups acting on manifolds: around the Zimmer program

TL;DR

This survey reviews the Zimmer program, which studies large group actions on compact manifolds, summarizing key questions, current progress, and open problems in the field.

Contribution

It provides a comprehensive overview of the Zimmer program, contextualizing its foundational questions, recent advances, and future research directions.

Findings

Summary of key conjectures and questions in the Zimmer program

Current state of research on group actions on manifolds

Identification of open problems and future directions

Abstract

This paper is a survey on the {\em Zimmer program}. In it's broadest form, this program seeks an understanding of actions of large groups on compact manifolds. The goals of this survey are $(1)$ to put in context the original questions and conjectures of Zimmer and Gromov that motivated the program, $(2)$ to indicate the current state of the art on as many of these conjectures and questions as possible and $(3)$ to indicate a wide variety of open problems and directions of research.