TITLE: projections of finitely presented groups
QUESTION [5 upvotes]: let's call an object $x$ of a cocomplete category (categorical) finitely generated if $\hom(x,-)$ commutes with filtered colimits of monomorphisms, and finitely presented if $\hom(x,-)$ even commutes with arbitrary filtered colimits. I know that in the literature there are some other definitions for these notions. at least in categories of algebraic structures, it would be nice that these notations coincide with the usual ones. let's concentrate on the category of groups, for these make problems ;).
it's easy to see that categorical finitely generated means the usual finitely generated. but I've figured out that a group $G$ is categorical finitely presented if and only if there is a finitely presented group $H$, over which $id_G : G \to G$ factors through. is this the same as beeing finitely presented?
the question is equivalent to: let $H$ be a finitely presented group and $p : H \to H$ a projection, i.e. a homomorphism satisfying $p^2 = p$. is then $im(p)$ also finitely presented?

REPLY [5 votes]: Here's a geometric proof that a retract of a finitely presented group is finitely presented.
This uses the fact that a f.g. group is coarsely simply connected iff it is finitely presented (Gromov calls this a triviality; it takes a little time to realize he's right).
Coarsely simply connected passes to retracts: it's clear because if you have a loop in the retract, coarse fill it in the all space and apply the retraction to the homotopy to get a homotopy inside the retract.
[The same argument shows that the Dehn function decreases when passing to retracts.]