Free factor complex

Short description: Concept in mathematics

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 $\mathrm{Out}(Fn)$.

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 ${{ℱ}}_n$ 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

${{ℱ}}_n^{(0)}=\{[A]|A\le F_n\text{ is a proper free factor of }{F}_n\}.$

(2) For $k\ge 1$, a $k$-simplex in ${{ℱ}}_n$ is a collection of $k+1$ distinct 0-cells $\{v_0,v_1,\dots ,v_k\}\subset {{ℱ}}_n^{(0)}$ 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 $A⪇B$.

For $n=2$ the above definition produces a complex with no $k$-cells of dimension $k\ge 1$. Therefore, ${{ℱ}}_2$ is defined slightly differently. One still defines ${{ℱ}}_2^{(0)}$ 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 $\{v_0,v_1\}\subset {{ℱ}}_2^{(0)}$ determine a 1-simplex in ${{ℱ}}_2$ if and only if there exists a free basis $a,b$ of $F_2$ such that $v_0=[⟨a⟩],v_1=[⟨b⟩]$. The complex ${{ℱ}}_2$ has no $k$-cells of dimension $k\ge 2$.

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

Main properties

• For every integer $n\ge 3$ the complex ${{ℱ}}_n$ is connected, locally infinite, and has dimension $n-2$. The complex ${{ℱ}}_2$ is connected, locally infinite, and has dimension 1.
• For $n=2$, the graph ${{ℱ}}_2$ is isomorphic to the Farey graph.
• There is a natural action of $\mathrm{Out}(Fn)$ on ${{ℱ}}_n$ by simplicial automorphisms. For a k-simplex $\mathrm{\Delta }=\{[A_0],\dots ,[A_k]\}$ and $\phi \in \mathrm{Out}(F_n)$ one has $\phi \mathrm{\Delta }:=\{[\phi (A_0)],\dots ,[\phi (A_k)]\}$.
• For $n\ge 3$ the complex ${{ℱ}}_n$ has the homotopy type of a wedge of spheres of dimension $n-2$.
• For every integer $n\ge 2$, the free factor graph ${{ℱ}}_n^{(1)}$, 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 ${{ℱ}}_n^{(1)}$, 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 $\phi \in \mathrm{Out}(F_n)$ acts as a loxodromic isometry of ${{ℱ}}_n^{(1)}$ if and only if $\phi$ is fully irreducible.
• There exists a coarsely Lipschitz coarsely $\mathrm{Out}(F_n)$-equivariant coarsely surjective map ${{ℱ}{𝒮}}_n\to {{ℱ}}_n^{(1)}$, where ${{ℱ}{𝒮}}_n$ 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 $\mathrm{Out}(F_n)$-equivariant coarsely surjective map $C{V}_n\to {{ℱ}}_n^{(1)}$, where $C{V}_n$ is the (volume-ones normalized) Culler–Vogtmann Outer space, equipped with the symmetric Lipschitz metric. The map $\pi$ takes a geodesic path in $C{V}_n$ to a path in ${ℱ}{F}_n$ contained in a uniform Hausdorff neighborhood of the geodesic with the same endpoints.
• The hyperbolic boundary $\partial {{ℱ}}_n^{(1)}$ of the free factor graph can be identified with the set of equivalence classes of "arational" $F_n$-trees in the boundary $\partial C{V}_n$ of the Outer space $C{V}_n$.
• The free factor complex is a key tool in studying the behavior of random walks on $\mathrm{Out}(F_n)$ and in identifying the Poisson boundary of $\mathrm{Out}(F_n)$.

Other models

There are several other models which produce graphs coarsely $\mathrm{Out}(Fn)$-equivariantly quasi-isometric to ${{ℱ}}_n^{(1)}$. These models include:

• The graph whose vertex set is ${{ℱ}}_n^0$ 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 ${𝒜},{ℬ}$ of $F_n$ such that $v_0=[{𝒜}],v_1=[{ℬ}]$ and ${𝒜}\cap {ℬ}\ne \varnothing$.

See also

• Mapping class group