Theorem. Every finitely presented subgroup of a direct product of two free groups is either free or contains a finite index subgroup of the form where and .
Central to the their proof is the construction of certain non-positively curved metric spaces whose fundamental groups are virtually the direct product of free groups. From there, it can be shown that finitely presented subgroups of the fundamental group exhibit the dichotomy given in the theorem. We begin by defining these spaces.
1. Square complexes
We will be working with square complexes. A square complex is a 2-dimensional CW complex in which every 2-cell is attached along a loop consisting of four (not necessarily distinct) 1-cells.
Every square complex can be metrized as a piecewise Euclidean metric space by treating each square as a Euclidean square. We wish to know when a square complex is non-positively curved. To do this we use the link graph for a vertex in . The vertices of this graph correspond to half-edges incident at , and two vertices are connected by an edge if these half-edges span a face in .
A VH-complex is a square complex in which every 1-cell can be labelled as either vertical or horizontal in such a manner so that the attaching loop of each 2-cell alternates between vertical and horizontal.
The labeling of the edges in a VH-complex induces a partition on the vertices in for each vertex . In particular this makes a bipartite graph. We will say a VH-complex is complete when is a complete bipartite graph for each . Intuitively, this means that if there is a vertical edge and a horizontal edge incident at a vertex then these edges must span a 2-cell in .
Complete VH-complexes are non-positively curved when their 2-cells are metrized as Euclidean squares. This is because all reduced circuits in a complete bipartite graph are of even length greater than two. Thus such a complex satisfies Gromov’s link condition.
3. The fundamental group of a complete VH-complex
We can think about VH-complexes in terms of gates and corridors. Let be the set of vertical edges in . A gate is a connected component of , while a corridor is a connected component of . We will say a VH-complex is clean if the attaching map of each corridor is injective.
To each VH-complex we can associate a graph whose vertices correspond to gates and whose edges correspond to corridors in . Note that this graph can have loops as a corridor may attach both ends to the same gate. When is clean and complete, the attaching maps of corridors are particularly nice. Ultimately, this allows one to see that has a covering space such that is a cover of . This leads to the following theorem from Wise’s PhD thesis:
Theorem 1 . Let be a complete VH-complex. If is clean and connected, then there exists a finite sheeted covering space which is a product of graphs.
A corollary brings us into the context of the direct product of two free groups.
Corollary. If is as above then for every vertex there exists a finite index subgroup of such that where consists of homotopy classes of loops in the vertical edges of and consists of homotopy classes of loops in the horizontal edges of . Furthermore, both and are free.
4. The lollipop dichotomy
A lollipop is a finite connected graph with exactly one vertex of valence one and exactly one vertex of valence three. A VH pair of lollipops is a 1-dimensional VH-complex whose vertical and horizontal subgraphs are lollipops and intersect a single vertex of valence two. A morphism from a VH pair of lollipops to a VH-complex is called locally geodesic if it is a local isometry. Locally geodesic VH pairs of lollipops are an obstruction to a VH-complex being complete. The following theorem applies to nice VH-complexes; this essentially means the complex is free of dangling edges and unnecessary faces.
Theorem 3. Let be a non-positively curved, compact VH-complex. If is nice then either is complete or contains a locally geodesic pair VH pair of lollipops.
A morphism from a pair of VH lollipops to a VH-complex which is -injective is called an essential pair of VH lollipops. The technical details of the following theorem require tower maps (see Bridson & Haeflinger), which the reader should think of as a akin to coverings.
Theorem 4. If is a complete VH-complex and is a finitely presented subgroup of then at least one of the following is true
- is a free group;
- there exists a complete VH-complex and a -injective tower max such that is conjugate to ;
- there is an essential pair of VH lollipops such that is conjugate to a subgroup of .
The hypotheses of the Corollary rule out option . If we assume is not free, the tower map allows us find a finite index subgroup of with the same sort of decomposition as in the Corollary. After arguing that a change of basepoint produces an isomorphism on the subgroups and , we come the last theorem we need.
Theorem 5. If is a clean, complete VH-complex and is a finitely presented subgroup of , then either is free or else contains a subgroup of finite index where consists of homotopy classes of vertical loops and consists of homotopy classes of horizontal loops.
The theorem quoted at the start of this post is a special case of the above theorem: it is the instance where is a product of two finite graphs with a single vertex.
As a concluding remark, we note that this style of argument doesn’t extend to subgroups of the direct product of more than two free groups as the tower maps argument breaks down for cubical complexes.