Groups not acting on manifolds

TL;DR

This paper investigates the limitations of group actions on compact manifolds, showing that most finitely generated groups cannot act smoothly while preserving volume, and introduces new examples of such groups.

Contribution

It demonstrates that generic finitely generated groups and many finitely presented torsion-free groups cannot act smoothly on compact manifolds.

Findings

Most finitely generated groups have no smooth volume-preserving actions on compact manifolds.

Constructs numerous finitely presented, torsion-free groups with no smooth actions.

Provides constraints on group actions on manifolds.

Abstract

In this article we collect a series of observations that constrain actions of many groups on compact manifolds. In particular, we show that "generic" finitely generated groups have no smooth volume preserving actions on compact manifolds while also producing many finitely presented, torsion free groups with the same property.