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 $A\leq G$ such that $A\neq \{1\},A\neq G$ and that there exists a subgroup $B\leq G$ such that $G=A\ast B$.

Let $n\geq 3$ 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

${\mathcal {F}}_{n}^{(0)}=\{[A]|A\leq F_{n}{\text{ is a proper free factor of }}F_{n}\}.$

(2) For $k\geq 1$, 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 $A_{0}\leq A_{1}\leq \dots \leq A_{k}$. [The assumption that these 0-cells are distinct implies that $A_{i}\neq A_{i+1}$ for $i=0,1,\dots ,k-1$]. In particular, a 1-cell is a collection $\{[A],[B]\}$ of two distinct 0-cells where $A,B\leq F_{n}$ 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 $k\geq 1$. 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 $k\geq 2$.

For $n\geq 2$ the 1-skeleton ${\mathcal {F}}_{n}^{(1)}$ is called the free factor graph for $F_{n}$.

Main properties

• For every integer $n\geq 3$ 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 $n\geq 3$ the complex ${\mathcal {F}}_{n}$ has the homotopy type of a wedge of spheres of dimension $n-2$.
• For every integer $n\geq 2$, 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 $n\geq 2$, 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 ${\mathcal {F}}F_{n}$ 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 ${\mathcal {A}}\cap {\mathcal {B}}\neq \varnothing$.

See also

• Mapping class group