Definition
The free factor complex, denoted FFₙ, is a simplicial complex associated to the free group Fₙ of rank n ( n ≥ 2). Its vertices are the conjugacy classes of proper non‑trivial free factors of Fₙ. A collection of k + 1 distinct vertices spans a k‑simplex precisely when the corresponding free factors can be realized as a chain under inclusion after possibly conjugating each factor; equivalently, there exist representatives A₀ ⊂ A₁ ⊂ ⋯ ⊂ A_k of the conjugacy classes with each A_i a proper free factor of Fₙ.
Overview
The free factor complex provides a combinatorial object on which the outer automorphism group Out(Fₙ) acts simplicially and with finite stabilizers. It plays a central role in the study of the geometry of Out(Fₙ), analogous to the curve complex for mapping class groups. The complex is infinite, locally infinite, and has dimension n − 2. Important results include:
- Hyperbolicity – Bestvina and Feighn (2014) proved that FFₙ is Gromov‑hyperbolic for all n ≥ 3.
- Connectivity – FFₙ is (n − 3)‑connected; in particular, it is simply connected for n ≥ 4.
- Cohomological dimension – The action of Out(Fₙ) on FFₙ yields bounds on the virtual cohomological dimension of Out(Fₙ).
These properties have made FFₙ a key tool in constructing analogues of Teichmüller theory for free groups and in proving rigidity and classification theorems for subgroups of Out(Fₙ).
Etymology / Origin
The term combines “free factor,” a standard notion in group theory denoting a subgroup A ≤ Fₙ such that Fₙ = A * B for some subgroup B, with “complex,” referring to a simplicial complex. The free factor complex was introduced in the late 1990s by Allen Hatcher and Karen Vogtmann in their work on the homology of outer automorphism groups of free groups (see Homology stability for outer automorphism groups of free groups, 1998). Subsequent detailed study, particularly its hyperbolic nature, was carried out by Mladen Bestvina and Mark Feighn.
Characteristics
| Feature | Description |
|---|---|
| Vertices | Conjugacy classes of proper non‑trivial free factors of Fₙ. |
| Simplices | Collections of vertices that can be represented by a chain of inclusions of free factors. |
| Dimension | n − 2 (the maximal length of a proper free factor chain is n − 1, giving a simplex of dimension n − 2). |
| Action | Out(Fₙ) acts by simplicial automorphisms; the action is cocompact but not proper (stabilizers are finite). |
| Topology | Connected; (n − 3)-connected; Gromov‑hyperbolic for n ≥ 3. |
| Metric | Usually equipped with the combinatorial metric where each edge has length 1, yielding the standard graph metric on its 1‑skeleton. |
| Relations | The link of a vertex is naturally identified with the free factor complex of the corresponding factor. The complex embeds equivariantly into the sphere complex of a related 3‑manifold model. |
Related Topics
- Outer space (CVₙ) – The contractible space of marked metric graphs on which Out(Fₙ) acts; FFₙ can be viewed as a boundary object for CVₙ.
- Free splitting complex – Another Out(Fₙ)‑invariant simplicial complex whose vertices correspond to free splittings of Fₙ.
- Curve complex – The analogous complex for surfaces; many techniques transfer between the two settings.
- Sphere complex – A complex of (isotopy classes of) essential spheres in a doubled handlebody, closely related to FFₙ via a natural map.
- Mapping class group – The group of isotopy classes of homeomorphisms of a surface; its action on the curve complex parallels the action of Out(Fₙ) on FFₙ.
The free factor complex continues to be an active area of research, particularly in connections to algorithmic problems in Out(Fₙ) and the study of subgroups with geometric or dynamical significance.