Free factor complex
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 Allen Hatcher and Karen 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(Fn)|].
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.
- For, the graph is isomorphic to the Farey graph.
- There is a natural action of [Out(Fn)|] 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 Mladen Bestvina and Mark 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(Fn)|]-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.