Free factor complex

In mathematics, the free factor complex (sometimes also called the complex of free factors) 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 $\operatorname{Out}(F_n)$.

Formal definition
For a free group $$G$$ a proper free factor of $$G$$ is a subgroup $$A\le G$$ such that $$A\ne \{1\}, A\ne G$$ and that there exists a subgroup $$ B\le G$$ such that $$G=A\ast B$$.

Let $$ n\ge 3$$ be an integer and let $$F_n$$ be the free group of rank $$n$$. The free factor complex $$ \mathcal F_n$$ for $$F_n$$ is a simplicial complex where:

(1) The 0-cells are the conjugacy classes in $$F_n$$ of proper free factors of $$F_n$$, that is
 * $$\mathcal F_n^{(0)}=\{[A] | A\le F_n \text{ is a proper free factor of } F_n \}.$$

(2) For $$k\ge 1$$, a $$k$$-simplex in $$ \mathcal F_n$$ is a collection of $$k+1$$ distinct 0-cells $$\{v_0, v_1, \dots, v_k\}\subset \mathcal F_n^{(0)}$$ such that there exist free factors $$A_0,A_1,\dots, A_k$$ of $$F_n$$ such that $$v_i=A_i$$ for $$i=0,1,\dots, k$$, and that $$A_0\le A_1\le \dots \le A_k$$. [The assumption that these 0-cells are distinct implies that $$A_i\ne A_{i+1}$$ for $$ i=0,1,\dots, k-1$$]. In particular, a 1-cell is a collection $$ \{[A], [B]\}$$ of two distinct 0-cells where $$A,B\le F_n$$ are proper free factors of $$F_n$$ such that $$A\lneq B$$.

For $$n=2$$ the above definition produces a complex with no $$k$$-cells of dimension $$k\ge 1$$. Therefore, $$\mathcal F_2$$ is defined slightly differently. One still defines $$\mathcal F_2^{(0)}$$ to be the set of conjugacy classes of proper free factors of $$F_2$$; (such free factors are necessarily infinite cyclic). Two distinct 0-simplices $$\{v_0,v_1\}\subset \mathcal F_2^{(0)}$$ determine a 1-simplex in $$\mathcal F_2$$ if and only if there exists a free basis $$a,b$$ of $$F_2$$ such that $$v_0=[\langle a\rangle], v_1=[\langle b\rangle]$$. The complex $$\mathcal F_2$$ has no $$k$$-cells of dimension $$k\ge 2$$.

For $$n\ge 2$$ the 1-skeleton $$\mathcal F_n^{(1)}$$ is called the free factor graph for $$F_n$$.

Main properties

 * For every integer $$n\ge 3$$ the complex $$\mathcal F_n$$ is connected, locally infinite, and has dimension $$n-2$$. The complex $$\mathcal F_2$$ is connected, locally infinite, and has dimension 1.
 * For $$n=2$$, the graph $$\mathcal F_2$$ is isomorphic to the Farey graph.
 * There is a natural action of $\operatorname{Out}(F_n)$ on $$\mathcal F_n$$ by simplicial automorphisms. For a k-simplex $$\Delta=\{[A_0],\dots, [A_k]\}$$ and $$\varphi\in \operatorname{Out}(F_n)$$ one has $$\varphi \Delta:=\{[\varphi(A_0)],\dots, [\varphi(A_k)]\}$$.
 * For $$n\ge 3$$ the complex $$\mathcal F_n$$ has the homotopy type of a wedge of spheres of dimension $$n-2$$.
 * For every integer $$n\ge 2$$, the free factor graph $$\mathcal F_n^{(1)}$$, equipped with the simplicial metric (where every edge has length 1), is a connected graph of infinite diameter.
 * For every integer $$n\ge 2$$, the free factor graph $$\mathcal F_n^{(1)}$$, 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 $$\varphi\in \operatorname{Out}(F_n)$$ acts as a loxodromic isometry of $$\mathcal F_n^{(1)}$$ if and only if $$\varphi$$ is fully irreducible.
 * There exists a coarsely Lipschitz coarsely $$\operatorname{Out}(F_n)$$-equivariant coarsely surjective map $$\mathcal{FS}_n\to \mathcal F_n^{(1)}$$, where $$\mathcal{FS}_n$$ 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 $$\operatorname{Out}(F_n)$$-equivariant coarsely surjective map $$CV_n\to \mathcal F_n^{(1)}$$, where $$CV_n$$ is the (volume-ones normalized) Culler–Vogtmann Outer space, equipped with the symmetric Lipschitz metric. The map $$\pi$$ takes a geodesic path in $$CV_n$$ to a path in $$\mathcal FF_n$$ contained in a uniform Hausdorff neighborhood of the geodesic with the same endpoints.
 * The hyperbolic boundary $$\partial \mathcal F_n^{(1)}$$ of the free factor graph can be identified with the set of equivalence classes of "arational" $$F_n$$-trees in the boundary $$\partial CV_n$$ of the Outer space $$CV_n$$.
 * The free factor complex is a key tool in studying the behavior of random walks on $$\operatorname{Out}(F_n)$$ and in identifying the Poisson boundary of $$\operatorname{Out}(F_n)$$.

Other models
There are several other models which produce graphs coarsely $\operatorname{Out}(F_n)$-equivariantly quasi-isometric to $$\mathcal F_n^{(1)}$$. These models include:


 * The graph whose vertex set is $$\mathcal F_n^{0}$$ and where two distinct vertices $$v_0,v_1$$ are adjacent if and only if there exists a free product decomposition $$F_n=A\ast B\ast C$$ such that $$v_0=[A]$$ and $$v_1=[B]$$.
 * The free bases graph whose vertex set is the set of $$F_n$$-conjugacy classes of free bases of $$F_n$$, and where two vertices $$v_0,v_1$$ are adjacent if and only if there exist free bases $$\mathcal A, \mathcal B$$ of $$F_n$$ such that $$v_0=[\mathcal A], v_1=[\mathcal B]$$ and $$\mathcal A\cap \mathcal B\ne \varnothing $$.