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 $\operatorname(F_n)$.

Formal definition

For a free group $G$ a ''proper free factor'' of $G$ is a subgroup $A\le G$ such that $A\ne \, A\ne G$ 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
:$\mathcal F_n^=\.$

(2) For $k\ge 1$, a $k$-simplex in $\mathcal F_n$ is a collection of $k+1$ distinct 0-cells $\\subset \mathcal F_n^$ 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 $A_0\le A_1\le \dots \le A_k$. he assumption that these 0-cells are distinct implies that $A_i\ne A_$ for $i=0,1,\dots, k-1$ In particular, a 1-cell is a collection $\$ 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 $\mathcal F_2^$ 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 $\\subset \mathcal F_2^$ 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/math>.
The complex $\mathcal F_2$ has no $k$-cells of dimension $k\ge 2$.

For $n\ge 2$ the 1-skeleton $\mathcal F_n^$ 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 $\operatorname(F_n)$ on $\mathcal F_n$ by simplicial automorphisms. For a ''k''-simplex $\Delta=\$ and $\varphi\in \operatorname(F_n)$ one has $\varphi \Delta:=\$.
*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 $\mathcal F_n^$, 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 $\mathcal F_n^$, 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 $\varphi\in \operatorname(F_n)$ acts as a loxodromic isometry of $\mathcal F_n^$ if and only if $\varphi$ is fully irreducible.
*There exists a coarsely Lipschitz coarsely $\operatorname(F_n)$-equivariant coarsely surjective map $\mathcal_n\to \mathcal F_n^$, where $\mathcal_n$ 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 $\operatorname(F_n)$-equivariant coarsely surjective map $CV_n\to \mathcal F_n^$, 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 $\partial \mathcal F_n^$ 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 $\operatorname(F_n)$ and in identifying the Poisson boundary of $\operatorname(F_n)$.

Other models

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

*The graph whose vertex set is $\mathcal F_n^$ 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= /math> and v_1= /math>.
*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/math> and $\mathcal A\cap \mathcal B\ne \varnothing$.

References

{{Reflist

See also

*Mapping class group

Geometric group theory
Geometric topology