# Is there a finitely generated residually finite group with solvable word problem that does not embed in a finitely presented residually finite group?

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?

• Same question seems to be also unclear in the category of f.g./f.p. linear groups. – Misha Jan 31 '14 at 5:40
• As far as I know it is not known whether the Grigorchuk group embeds into a f.p. residually finite group. – Mustafa Gokhan Benli Jan 31 '14 at 6:11
• I am less optimistic about linear groups. – Benjamin Steinberg Jan 31 '14 at 16:04
• @Misha, Benjamin: Benjamin you're right: a countable infinite simple linear group such as $PSL_2(\mathbf{Q})$ can't be embedded into a f.g. linear group, because the former is not residually finite. – YCor Feb 10 '14 at 10:38
• @YvesCornulier: True, but I was thinking about finitely-generated groups (embedding f.g linear groups in f.p. linear groups). (I guess, I was not very clear about it.) I do not see any obstructions for this. – Misha Feb 10 '14 at 13:50