  Working in the framework of Borel reducibility, we study various notions of
embeddability between groups. We prove that the embeddability between countable
groups, the topological embeddability between (discrete) Polish groups, and the
isometric embeddability between separable groups with a bounded bi-invariant
complete metric are all invariantly universal analytic quasi-orders. This
strengthens some results from [Wil14] and [FLR09].
