In mathematics, the free factor complex is a free group counterpart of the notion of the curve complex of a finite type surface. The free factor complex was originally introduced in a 1998 paper of Hatcher and Vogtmann. Like the curve complex, the free factor complex is known to be Gromov-hyperbolic. The free factor complex plays a significant role in the study of large-scale geometry of Out|.
Formal definition
For a free group a proper free factor of is a subgroup such that and that there exists a subgroup such that. Let be an integer and let be the free group of rank. The free factor complex for is a simplicial complex where: The 0-cells are the conjugacy classes in of proper free factors of, that is For, a -simplex in is a collection of distinct 0-cells such that there exist free factors of such that for, and that. . In particular, a 1-cell is a collection of two distinct 0-cells where are proper free factors of such that. For the above definition produces a complex with no -cells of dimension. Therefore, is defined slightly differently. One still defines to be the set of conjugacy classes of proper free factors of ;. Two distinct 0-simplices determine a 1-simplex in if and only if there exists a free basis of such that. The complex has no -cells of dimension. For the 1-skeleton is called the free factor graph for.
Main properties
For every integer the complex is connected, locally infinite, and has dimension. The complex is connected, locally infinite, and has dimension 1.
There is a natural action of Out| on by simplicial automorphisms. For a k-simplex and one has.
For the complex has the homotopy type of a wedge of spheres of dimension.
For every integer, the free factor graph, equipped with the simplicial metric, is a connected graph of infinite diameter.
For every integer, the free factor graph, equipped with the simplicial metric, is Gromov-hyperbolic. This result was originally established by Bestvina and Feighn; see also for subsequent alternative proofs.
An element acts as a loxodromic isometry of if and only if is fully irreducible.
There exists a coarsely Lipschitz coarsely -equivariant coarsely surjective map, where is the free splittings complex. However, this map is not a quasi-isometry. The free splitting complex is also known to be Gromov-hyperbolic, as was proved by Handel and Mosher.
Similarly, there exists a natural coarsely Lipschitz coarsely -equivariant coarsely surjective map, where is the Culler–Vogtmann Outer space, equipped with the symmetric Lipschitz metric. The map takes a geodesic path in to a path in contained in a uniform Hausdorff neighborhood of the geodesic with the same endpoints.
The hyperbolic boundary of the free factor graph can be identified with the set of equivalence classes of "arational" -trees in the boundary of the Outer space.
The free factor complex is a key tool in studying the behavior of random walks on and in identifying the Poisson boundary of.
Other models
There are several other models which produce graphs coarsely Out|-equivariantly quasi-isometric to. These models include:
The graph whose vertex set is and where two distinct vertices are adjacent if and only if there exists a free product decomposition such that and.
The free bases graph whose vertex set is the set of -conjugacy classes of free bases of, and where two vertices are adjacent if and only if there exist free bases of such that and.
