Our aim is to prove Howson’s theorem for free groups followings Stallings’ article, Topology of finite graphs; this note is more or less a sequel of How Stallings proved finitely-generated free groups are subgroup separable. The construction introduced here, the fiber product, will be also useful for the generalization of ideas of Stallings to special cube complexes (due to Wise).
Another interesting proof, based on regular languages, can be found in Meier’s book untitled Graphs, groups and trees.
Theorem: (Howson) Let be a free group of finite rank and be two finitely-generated subgroups. Then is also finitely-generated.
Let and be two immersions (ie. locally injective cellular maps) between graphs. We define the fiber product as the graph whose vertices are
where two vertices and are linked by an edge if and only if and are linked by an edge in .
Notice that there are two obvious projections , , and a natural map so that we have the following commutative diagram:
Claim 1: is an immersion.
Proof. Let be a vertex of and two edges starting from ; let and denote the ending points of and . If , then hence . Therefore, if then and are two loops based at , sent via on two loops based at . So is locally injective.
Claim 2: .
First, we need the following easy lemma:
Lemma: Let be a graph and be two reduced (ie. without round-trip) loops. If in then .
Sketch of proof. The result is clear if is a bouquet of circles, using the usual normal form for free groups. In fact, it is the only case to consider, since the quotient of by a maximal subtree is a bouquet of circles.
Proof of claim 2. Because , the inclusion
is clear. Conversely, let , that is there exist and such that in . Of course, and may be chosen reduced so that and so are. We deduce that from our previous lemma. Therefore, by construction of , there exist a loop such that , hence the inclusion
Proof of Howson’s theorem. As in the previous note, we see as the fundamental group of a finite bouquet of circles , and we choose immersions of finite graphs and such that and . Then the fiber product gives an immersion such that ; because is a finite graph (since and are itself finite), we deduce that is finitely-generated.
See also Graphs and Free Groups.