# Is every finitely generated group colimit of residually finite groups

I was listening to a talk about ultraproducts and one result there suggested, that every finitely generated group can be written as a colimit of residually finite groups (over a directed system). As I don't see a trivial proof, I expect it to be false (it is a statement about all groups). But I don't see a counterexample.

• For context, these are called LEF groups. LEF means "locally embeddable in finite" (groups). Such a notion makes sense for much more general algebraic structures, see Malcev's book "algebraic systems"; in the context of groups it was later specified by Stëpin, and then Vershik-Gordon. For finitely presented groups, LEF is equivalent to residually finite. – YCor Oct 22 '18 at 12:01

$$H:= \langle a,b,c,d \mid ba = a^2b, cb=b^2c, dc=c^2d, ad=d^2a \rangle.$$