The famous Higman embedding theorem says that every recursively presented group embeds in a finitely presented group. This is a convenient tool to construct finitely presented groups with bizarre properties from recursively presented ones, which are usually easier to construct.

One cannot hope for an exact analogue of Higman's theorem in the setting of residually finite groups because finitely presented residually finite groups have solvable word problem and hence their finitely generated subgroups do as well. But Meskin constructed finitely generated recursively presented groups with undecidable word problem. I know of no other obstruction to embedding a finitely generated residually finite group into a finitely presented one, so I ask the following question.

Does there exist a finitely generated residually finite group with decidable word problem that cannot be embedded in a finitely presented residually finite group?

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