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 [math]\displaystyle{ \operatorname{Out}(F n) }[/math].

Formal definition

For a free group [math]\displaystyle{ G }[/math] a proper free factor of [math]\displaystyle{ G }[/math] is a subgroup [math]\displaystyle{ A\le G }[/math] such that [math]\displaystyle{ A\ne \{1\}, A\ne G }[/math] and that there exists a subgroup [math]\displaystyle{ B\le G }[/math] such that [math]\displaystyle{ G=A\ast B }[/math].

Let [math]\displaystyle{ n\ge 3 }[/math] be an integer and let [math]\displaystyle{ F_n }[/math] be the free group of rank [math]\displaystyle{ n }[/math]. The free factor complex [math]\displaystyle{ \mathcal F_n }[/math] for [math]\displaystyle{ F_n }[/math] is a simplicial complex where:

(1) The 0-cells are the conjugacy classes in [math]\displaystyle{ F_n }[/math] of proper free factors of [math]\displaystyle{ F_n }[/math], that is

[math]\displaystyle{ \mathcal F_n^{(0)}=\{[A] | A\le F_n \text{ is a proper free factor of } F_n \}. }[/math]

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

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

For [math]\displaystyle{ n\ge 2 }[/math] the 1-skeleton [math]\displaystyle{ \mathcal F_n^{(1)} }[/math] is called the free factor graph for [math]\displaystyle{ F_n }[/math].

Main properties

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

Other models

There are several other models which produce graphs coarsely [math]\displaystyle{ \operatorname{Out}(F n) }[/math]-equivariantly quasi-isometric to [math]\displaystyle{ \mathcal F_n^{(1)} }[/math]. These models include:

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

See also

• Mapping class group