Simplicity of the group of compactly supported area preserving homeomorphisms of the open disc and fragmentation of symplectic diffeomorphisms

TL;DR

This paper explores the simplicity of the group of area-preserving homeomorphisms of the 2-disc, linking it to a fragmentation property of symplectic diffeomorphisms, and establishing conditions for group simplicity.

Contribution

It establishes an equivalence between the simplicity of the homeomorphism group and a specific fragmentation property of symplectic diffeomorphisms.

Findings

Simplicity of the homeomorphism group is equivalent to a fragmentation property.

Existence of a uniform bound m for decomposing elements supported on discs.

Connection between topological and symplectic group properties.

Abstract

In 1980, Albert Fathi asked whether the group of area-preserving homeomorphisms of the 2-disc that are the identity near the boundary is a simple group. In this paper, we show that the simplicity of this group is equivalent to the following fragmentation property in the group of compactly supported, area preserving diffeomorphisms of the plane: there exists a constant m such that every element supported on a disc D is the product of at most m elements supported on topological discs whose area are half the area of D.