Is every group isomorphic to an inductive colimit (that is, directed colimit, also called inductive limit, or directed limit) of free groups?


I guess, the answer is no. In that case: Is there a characterization of the groups that are isomorphic to inductive colimits of free groups?



Inductive colimits of free groups
Let $G$ be a group. Equivalences:


1. $G$ is locally free (i.e., all its finitely generated subgroups are free)
2. $G$ is a filtering inductive limit of free groups with injective connecting homomorphisms
3. $G$ is a filtering inductive limit of free groups.


The equivalence between (1) and (2) is clear (given that subgroups of free groups are free - otherwise "locally free" would be a bit imprecise); (3) is tautologically implied by (2).


Conversely suppose (3), and let's prove (1). Passing to a finitely generated subgroup of $G$ reduces to $G$ finitely generated. Then passing to a cofinal subsystem allows to assume that the filtering system has an initial element $H$ with $H\to G$ surjective. Then some finitely generated subgroup $K$ of $H$ has onto image in $G$ and then replacing each group in the filtering system, we realize $G$ as a filtering inductive limit of finitely generated free groups with surjective connecting homomorphisms. Then the rank of these free groups decreases until it stabilizes, and since finitely generated free groups are Hopfian, this means that there's a cofinal subsystem made up of isomorphisms. So $G$ is free.


Note: for more other classes of groups, usually such an implication as (3)$\Rightarrow$(2) fails. For instance, every group is a filtering inductive limit of finitely presented groups (this is not specific to groups!), but usually this cannot be done with injective connecting homomorphisms (e.g., for an infinitely presented finitely generated group).


