On two remarkable groups of area-preserving homeomorphisms

TL;DR

This paper investigates the structure of groups of area-preserving homeomorphisms on a symplectic sphere, revealing a proper subgroup and identifying infinite-dimensional flats within their quotient space.

Contribution

It establishes that the Hamiltonian homeomorphisms form a proper normal subgroup of finite energy Hamiltonian homeomorphisms and uncovers infinite-dimensional flats in their Hofer metric quotient.

Findings

Hamiltonian homeomorphisms form a proper normal subgroup

Existence of infinite-dimensional flats in the quotient space

Insights into the structure of symplectic homeomorphism groups

Abstract

We prove that on a symplectic sphere, the group of Hamiltonian homeomorphisms in the sense of Oh and M\"uller is a proper normal subgroup of the group of finite energy Hamiltonian homeomorphisms. Moreover we detect infinite-dimensional flats inside the quotient of these groups endowed with the natural Hofer pseudo-metric.