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

Many transformation groups of manifolds are simple, but their universal coverings are not. In this 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 [Formula: see text] of the group of Hamiltonian diffeomorphisms of a closed symplectic manifold [Formula: see text]. Tsuboi constructed a metric space [Formula: see text] for a simple group [Formula: see text]. We generalize his construction to relatively simple groups, and study their large-scale geometric structure. In particular, Tsuboi’s metric space of [Formula: see text] is not quasi-isometric to the half-line for every closed symplectic manifold [Formula: see text].