Essential subgroups of topological groups

We study the class W of Hausdorff topological groups G for which the follow ing two cardinal invariants coincide: ES(G) = min{/H/ : H less than or equal to G dense and essential} TD(G) = min(/H/ : H less than or equal to G totally dense). We prove that W contains the: following classes: locally compact abelian gr oups, compact connected groups, countably compact totally disconnected abel ian groups, topologically simple groups, locally compact Abelian groups whe n endowed with their Bohr topology, totally minimal abelian groups and free Abelian topological groups. For all these classes we are also able to ive an explicit computation of the common value of ES arid TD.