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

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 $(M,\omega)$.