Fact: There exists a finitely generated non-Hopf group.  (An example is the Baumslag-Solitar group B(2,3), although we cannot prove it yet.)  So, by Lemma 5, there is a finitely generated non-residually finite group.  Thus, free groups are not ERF: if G is a finitely generated non-residually finite group, then Lemma 4 implies that the kernel of a surjection F_r \rightarrow G is not separable in F_r.  However, finitely generated subgroups of free groups are separable:

Marshall Hall’s Theorem (1949): F_r is LERF.

This proof is associated with Stallings.

Proof: As usual, let F_r = \pi_1(X) where X is a rose. Let X' \rightarrow X be a covering map with \pi_1 (X') finitely generated.  Let \Delta \subset X' be compact. We need to embed \Delta in an intermediate finite-sheeted covering.

Enlarging \Delta if necessary, we may assume that \Delta is connected and that \pi_1 (\Delta) = \pi_1(X').  Note that we have \Delta \subset X' \rightarrow X. By Theorem 5 (see below), the immersion \Delta \rightarrow X extends to a covering \hat{X} \rightarrow X. Then \pi_1 (X') = \pi_1(\Delta) \subset \pi_1 (\hat{X}). So X' \rightarrow X lifts to a map X' \rightarrow \hat{X}.

The main tool in the proof above is this:

Theorem 5: The immersion \Delta \rightarrow X can be completed to a finite-sheeted covering \hat{X} \rightarrow X into which \Delta embeds:


Proof: Color and orient the edges of X. Any combinatorial map of graphs \Delta \rightarrow X corresponds uniquely to a coloring and orientation on the edges of \Delta.  A combinatorial map is an immersion if and only if at every vertex of \Delta, we see each color arriving at most once and leaving at most once. Likewise, it’s a covering map if and only if at each vertex, we see each color arriving exactly once and leaving exactly once.

Completing $latex \Delta$ to a finite-sheeted covering $latex \hat{X} \rightarrow X$.

Let k be the number of vertices of \Delta.  For each color c, let k_c be the number of edges of \Delta colored c. Then there are k-k_c vertices of \Delta missing “arriving” edges colored c, and there are k-k_c vertices of \Delta missing “leaving” edges colored c.  Choose any bijection between these two sets and use this to glue in k-k_c edges colored c. When this is done for all colors, the resulting map \Delta \subset \hat{X} \rightarrow X is clearly a covering.

Note that the proof in fact gives us more.  For instance:

Exercise 6: If H is a finitely generated subgroup of F_r, then H is a free factor of a finite-index subgroup of F_r.

Exercise 7 (Greenberg’s Theorem): If H\triangleleft F_r and H is finitely generated, then H is of finite index in F_r.