On Jordan type bounds for finite groups of diffeomorphisms of 3-manifolds and Euclidean spaces

TL;DR

This paper investigates whether finite subgroups of diffeomorphism groups of manifolds have bounded abelian subgroups, confirming this for all compact 3-manifolds and Euclidean spaces under dimension constraints, extending classical Jordan bounds.

Contribution

It extends Jordan's classical bounds to finite subgroups of diffeomorphism groups for all compact 3-manifolds and Euclidean spaces of dimension less than 7.

Findings

Bounded abelian subgroups exist for finite diffeomorphism groups of all compact 3-manifolds.

Bounded abelian subgroups exist for Euclidean spaces of dimension less than 7.

Open problems remain for higher-dimensional spheres and Euclidean spaces.

Abstract

By a classical result of Jordan, each finite subgroup G of a complex linear group GL_n(C) has an abelian subgroup whose index in G is bounded by a constant depending only on n. We consider the problem if this remains true for finite subgroups G of the diffeomorphism group of a smooth manifold, and show that it is true for all compact 3-manifolds as well as for Euclidean spaces of dimension n < 7. The question remains open at present e.g. for odd-dimensional spheres of dimension greater or equal to five, and for Euclidean spaces of dimension greater or equal to seven.