2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/77634If a class of finitely generated groups Curly(G) is closed under isometric amalgamations along free subgroups, then every G in Curly(G) can be quasi-isometrically embedded in a group Hat(G) in Curly(G) that has no proper subgroups of finite index. Every compact, connected, non-positively curved space X admits an isometric embedding into a compact, connected, non-positively curved space Overline(X) such that Overline(X) has no non-trivial finite-sheeted coverings.18 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTMon1/paper4.abs.htmlGroup TheoryGeometric Topology20E26, 20E06, 53C70, 20F32, 20F06text