I think the magic acronym is CEP. See http://en.wikipedia.org/wiki/CEP_subgroup
Update I've just noticed that the question is also tagged reference-request, so here's one (Google reveals many, I'm not sure if there is a canonical one): Non-amenable finitely presented torsion-by-cyclic groups by Ol'shanskii and Sapir. See the introductory Section 1.2 on Page 3 of the pdf.