Proper actions of discrete groups of affine transformations

TL;DR

This paper reviews the historical background, recent progress, and future directions concerning proper affine actions of discrete groups, especially highlighting Margulis's groundbreaking 1980s work on free groups in 3-space.

Contribution

It synthesizes developments in the study of proper affine actions, emphasizing Margulis's pioneering results and outlining open problems and research directions.

Findings

Margulis proved proper affine actions of free groups in 3-space.

The paper discusses recent advances in understanding affine group actions.

Future research directions are proposed based on historical and recent developments.

Abstract

In the early 1980's Margulis startled the world by showing the existence of proper affine actions of free groups on 3-space, answering a provocative and suggestive question Milnor posed in 1977. In this paper we discuss the historical background motivating this question, recent progress on this subject, and future directions inspired by this discovery.