Free factor complex

Template:Short description 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.^[1] 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 $Out (F_{n})$ .

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 $G = A * B$ .

Let $n \geq 3$ 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 \leq F_{n} is a proper free factor of F_{n}} .

(2) For $k \geq 1$ , 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 $A_{0} \leq A_{1} \leq \dots \leq A_{k}$ . [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 $A ⪇ B$ .

For $n = 2$ the above definition produces a complex with no $k$ -cells of dimension $k \geq 1$ . 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 $k \geq 2$ .

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

Main properties

For every integer $n \geq 3$ 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 $Out (F_{n})$ on $ℱ_{n}$ by simplicial automorphisms. For a k-simplex $Δ = {[A_{0}], \dots, [A_{k}]}$ and $φ \in Out (F_{n})$ one has $φ Δ := {[φ (A_{0})], \dots, [φ (A_{k})]}$ .
For $n \geq 3$ the complex $ℱ_{n}$ has the homotopy type of a wedge of spheres of dimension $n - 2$ .^[1]
For every integer $n \geq 2$ , 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.^[2]^[3]
For every integer $n \geq 2$ , 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;^[4] see also ^[5]^[6] for subsequent alternative proofs.
An element $φ \in Out (F_{n})$ acts as a loxodromic isometry of $ℱ_{n}^{(1)}$ if and only if $φ$ is fully irreducible.^[4]
There exists a coarsely Lipschitz coarsely $Out (F_{n})$ -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.^[7]
Similarly, there exists a natural coarsely Lipschitz coarsely $Out (F_{n})$ -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 $π$ 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.^[4]
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 $C V_{n}$ .^[8]
The free factor complex is a key tool in studying the behavior of random walks on $Out (F_{n})$ and in identifying the Poisson boundary of $Out (F_{n})$ .^[9]

Other models

There are several other models which produce graphs coarsely $Out (F_{n})$ -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 $F_{n} = A * B * C$ 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 $𝒜 \cap ℬ \neq \emptyset$ .^[5]

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Free factor complex

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Formal definition

Main properties

Other models

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Free factor complex

Formal definition

Main properties

Other models

References

See also

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