# Free factor complex

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In mathematics, the **free factor complex** (sometimes also called the **complex of free factors**) is a [free group](/source/Free_group) counterpart of the notion of the [curve complex](/source/Curve_complex) of a finite type surface. The free factor complex was originally introduced in a 1998 paper of [Allen Hatcher](/source/Allen_Hatcher) and [Karen Vogtmann](/source/Karen_Vogtmann).[1] Like the curve complex, the free factor complex is known to be [Gromov-hyperbolic](/source/%CE%94-hyperbolic_space). The free factor complex plays a significant role in the study of large-scale geometry of [\operatorname{Out}(F_n)](/source/Out(Fn)).

## Formal definition

For a free group G a *proper free factor* of G is a [subgroup](/source/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](/source/Free_group) of rank n. The **free factor complex** \mathcal F_n for F_n is a [simplicial complex](/source/Simplicial_complex) where:

(1) The 0-cells are the [conjugacy classes](/source/Conjugacy_class) in F_n of proper free factors of F_n, that is

- \mathcal 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 \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\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\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^{(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\ge 2.

For n\ge 2 the 1-skeleton \mathcal F_n^{(1)} 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](/source/Farey_graph).
- There is a natural [action](/source/Group_action_(mathematics)) of [\operatorname{Out}(F_n)](/source/Out(Fn)) 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\ge 3 the complex \mathcal F_n has the [homotopy type](/source/Homotopy_type) of a wedge of spheres of dimension n-2.[1]
- For every integer n\ge 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.[2][3]
- For every integer n\ge 2, the free factor graph \mathcal F_n^{(1)}, equipped with the simplicial metric, is [Gromov-hyperbolic](/source/%CE%94-hyperbolic_space). This result was originally established by [Mladen Bestvina](/source/Mladen_Bestvina) and Mark Feighn;[4] see also [5][6] 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](/source/Fully_irreducible_automorphism).[4]
- 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](/source/Free_splittings_complex). However, this map is not a [quasi-isometry](/source/Quasi-isometry). The free splitting complex is also known to be [Gromov-hyperbolic](/source/%CE%94-hyperbolic_space), as was proved by Handel and Mosher.[7]
- 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](/source/Outer_space_(mathematics)), 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.[4]
- 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.[8]
- The free factor complex is a key tool in studying the behavior of [random walks](/source/Random_walk) on \operatorname{Out}(F_n) and in identifying the [Poisson boundary](/source/Poisson_boundary) of \operatorname{Out}(F_n).[9]

## Other models

There are several other models which produce graphs coarsely [\operatorname{Out}(F_n)](/source/Out(Fn))-equivariantly [quasi-isometric](/source/Quasi-isometry) 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\ne \varnothing.[5]

## References

1. Hatcher, Allen & Vogtmann, Karen (1998). "The complex of free factors of a free group". *[Quarterly Journal of Mathematics](/source/Quarterly_Journal_of_Mathematics)*. **49** (196): 459–468. Series 2. [arXiv:2203.15602](https://arxiv.org/abs/2203.15602). [doi:10.1093/qmathj/49.4.459](https://doi.org/10.1093/qmathj/49.4.459)

1. Kapovich, Ilya & Lustig, Martin (2009). "Geometric intersection number and analogues of the curve complex for free groups". *[Geometry & Topology](/source/Geometry_%26_Topology)*. **13** (3): 1805–1833. [arXiv:0711.3806](https://arxiv.org/abs/0711.3806). [doi:10.2140/gt.2009.13.1805](https://doi.org/10.2140/gt.2009.13.1805)

1. Behrstock, Jason; Bestvina, Mladen; Clay, Matt (2010). "Growth of intersection numbers for free group automorphisms". *[Journal of Topology](/source/Journal_of_Topology)*. **3** (2): 280–310. [arXiv:0806.4975](https://arxiv.org/abs/0806.4975). [doi:10.1112/jtopol/jtq008](https://doi.org/10.1112/jtopol/jtq008)

1. Bestvina, Mladen & Feighn, Mark (2014). "Hyperbolicity of the complex of free factors". *[Advances in Mathematics](/source/Advances_in_Mathematics)*. **256**: 104–155. [arXiv:1107.3308](https://arxiv.org/abs/1107.3308). [doi:10.1016/j.aim.2014.02.001](https://doi.org/10.1016/j.aim.2014.02.001)

1. Kapovich, Ilya & Rafi, Kasra (2014). "On hyperbolicity of free splitting and free factor complexes". *[Groups, Geometry, and Dynamics](/source/Groups,_Geometry,_and_Dynamics)*. **8** (2): 391–414. [arXiv:1206.3626](https://arxiv.org/abs/1206.3626). [doi:10.4171/GGD/231](https://doi.org/10.4171/GGD/231)

1. Hilion, Arnaud & Horbez, Camille (2017). "The hyperbolicity of the sphere complex via surgery paths". *[Journal für die reine und angewandte Mathematik](/source/Journal_f%C3%BCr_die_reine_und_angewandte_Mathematik)*. **730**: 135–161. [arXiv:1210.6183](https://arxiv.org/abs/1210.6183). [doi:10.1515/crelle-2014-0128](https://doi.org/10.1515/crelle-2014-0128)

1. Handel, Michael & Mosher, Lee (2013). "The free splitting complex of a free group, I: hyperbolicity". *[Geometry & Topology](/source/Geometry_%26_Topology)*. **17** (3): 1581–1672. [arXiv:1111.1994](https://arxiv.org/abs/1111.1994). [doi:10.2140/gt.2013.17.1581](https://doi.org/10.2140/gt.2013.17.1581). MR 3073931.

1. Bestvina, Mladen & Reynolds, Patrick (2015). "The boundary of the complex of free factors". *[Duke Mathematical Journal](/source/Duke_Mathematical_Journal)*. **164** (11): 2213–2251. [arXiv:1211.3608](https://arxiv.org/abs/1211.3608). [doi:10.1215/00127094-3129702](https://doi.org/10.1215/00127094-3129702)

1. Horbez, Camille (2016). "The Poisson boundary of \operatorname{Out}(F_N)". *[Duke Mathematical Journal](/source/Duke_Mathematical_Journal)*. **165** (2): 341–369. [arXiv:1405.7938](https://arxiv.org/abs/1405.7938). [doi:10.1215/00127094-3166308](https://doi.org/10.1215/00127094-3166308)

## See also

- [Mapping class group](/source/Mapping_class_group)

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