Relative simplicity of the universal coverings of transformation groups and Tsuboi's metric

TL;DR

This paper explores the concept of relatively simple groups, showing many universal coverings of transformation groups are relatively simple, and generalizes Tsuboi's metric space construction to these groups, revealing their large-scale geometry.

Contribution

It introduces the notion of relatively simple groups, demonstrates many universal coverings are relatively simple, and extends Tsuboi's metric space framework to analyze their geometric properties.

Findings

Universal coverings of many transformation groups are relatively simple.

Tsuboi's metric space for these groups is not quasi-isometric to the half line.

The large-scale geometry of these metric spaces is studied and characterized.

Abstract

Many transformation groups on manifolds are simple, but their universal coverings are not. In the present paper, we study the concept of relatively simple group, that is, a group with the maximum proper normal subgroup. We show that many examples of universal coverings of transformation groups are relatively simple, including the universal covering $\widetilde{\mathrm{Ham}}(M,\omega)$ of the group of Hamiltonian diffeomorphisms of a closed symplectic manifold $(M,\omega)$. Tsuboi constructed a metric space $\mathcal{M}(G)$ for a simple group $G$. We generalize his construction to relatively simple groups, and study their large scale geometric structure. In particular, Tsuboi's metric space of $\widetilde{\mathrm{Ham}}(M, \omega)$ is not quasi-isometric to the half line for every closed symplectic manifold $(M,\omega)$.