Malnormal Subgroups of Finitely Presented Groups

The following refinement of the Higman embedding theorem is proved: Given a finitely generated recursively presented group R, there exists a quasi-isometric malnormal embedding of R into a finitely presented group H such that the image of the embedding enjoys the Congruence Extension Property. Moreover, it is shown that the group H can be constructed to have decidable Word problem if and only if the Word problem of R is decidable, yielding a refinement of a theorem of Clapham. Finally, it is proved that for any countable group G and any computable function :G satisfying some necessary requirements, there exists a malnormal embedding enjoying the Congruence Extension Property of G into a finitely presented group H such that the restriction of |·|H to G is equivalent to , producing a refinement of a result of Ol'shanskii.