Fragmented Hofer's geometry on Hameomorphism groups and its application

TL;DR

This paper investigates the structure of Hamiltonian homeomorphism groups using fragmented Hofer's norms, showing they are not simple if M"uller’s question about Hofer's metrics equivalence is affirmatively answered.

Contribution

It introduces fragmented Hofer's norms on Hamiltonian homeomorphism groups and links their properties to the group's simplicity, addressing M"uller's open problem.

Findings

Hamiltonian homeomorphism group is not simple under certain conditions

Fragmented Hofer's norms are effective tools for group analysis

Addresses M"uller's question on Hofer's metrics equivalence

Abstract

Stefan M$\ddot{\mathrm{u}}$ller posed the problem "Do Hofer's metrics on the group of Hamiltonian diffeomorphism and the one of Hamiltonian homeomorphisms (Hameomorphisms) correspond?". Let $(M,\omega)$ be a compact exact symplectic manifold. We prove that the group of Hamiltonian homeomorphisms is not a simple group if the positive answer of M$\ddot{\mathrm{u}}$ller's question holds . To prove this results, we introduce fragmented Hofer's norms on a normal subgroup of the group of Hamiltonian homeomorphisms.