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 Hatcher and 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 ${\displaystyle \operatorname {Out} (F_{n})}$.

Formal definition

For a free group ${\displaystyle G}$ a proper free factor of ${\displaystyle G}$ is a subgroup ${\displaystyle A\leq G}$ such that ${\displaystyle A\neq \{1\},A\neq G}$ and that there exists a subgroup ${\displaystyle B\leq G}$ such that ${\displaystyle G=A\ast B}$.

Let ${\displaystyle n\geq 3}$ be an integer and let ${\displaystyle F_{n}}$ be the free group of rank ${\displaystyle n}$. The free factor complex ${\displaystyle {\mathcal {F}}_{n}}$ for ${\displaystyle F_{n}}$ is a simplicial complex where:

(1) The 0-cells are the conjugacy classes in ${\displaystyle F_{n}}$ of proper free factors of ${\displaystyle F_{n}}$, that is

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

(2) For ${\displaystyle k\geq 1}$, a ${\displaystyle k}$-simplex in ${\displaystyle {\mathcal {F}}_{n}}$ is a collection of ${\displaystyle k+1}$ distinct 0-cells ${\displaystyle \{v_{0},v_{1},\dots ,v_{k}\}\subset {\mathcal {F}}_{n}^{(0)}}$ such that there exist free factors ${\displaystyle A_{0},A_{1},\dots ,A_{k}}$ of ${\displaystyle F_{n}}$ such that ${\displaystyle v_{i}=A_{i}}$ for ${\displaystyle i=0,1,\dots ,k}$, and that ${\displaystyle A_{0}\leq A_{1}\leq \dots \leq A_{k}}$. [The assumption that these 0-cells are distinct implies that ${\displaystyle A_{i}\neq A_{i+1}}$ for ${\displaystyle i=0,1,\dots ,k-1}$]. In particular, a 1-cell is a collection ${\displaystyle \{[A],[B]\}}$ of two distinct 0-cells where ${\displaystyle A,B\leq F_{n}}$ are proper free factors of ${\displaystyle F_{n}}$ such that ${\displaystyle A\lneq B}$.

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

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

Main properties

• For every integer ${\displaystyle n\geq 3}$ the complex ${\displaystyle {\mathcal {F}}_{n}}$ is connected, locally infinite, and has dimension ${\displaystyle n-2}$. The complex ${\displaystyle {\mathcal {F}}_{2}}$ is connected, locally infinite, and has dimension 1.
• For ${\displaystyle n=2}$, the graph ${\displaystyle {\mathcal {F}}_{2}}$ is isomorphic to the Farey graph.
• There is a natural action of ${\displaystyle \operatorname {Out} (F_{n})}$ on ${\displaystyle {\mathcal {F}}_{n}}$ by simplicial automorphisms. For a k-simplex ${\displaystyle \Delta =\{[A_{0}],\dots ,[A_{k}]\}}$ and ${\displaystyle \varphi \in \operatorname {Out} (F_{n})}$ one has ${\displaystyle \varphi \Delta$ :=\{[\varphi (A_{0})],\dots ,[\varphi (A_{k})]\}} .
• For ${\displaystyle n\geq 3}$ the complex ${\displaystyle {\mathcal {F}}_{n}}$ has the homotopy type of a wedge of spheres of dimension ${\displaystyle n-2}$.[1]
• For every integer ${\displaystyle n\geq 2}$, the free factor graph ${\displaystyle {\mathcal {F}}_{n}^{(1)}}$, equipped with the simplicial metric (where every edge has length 1), is a connected graph of infinite diameter.[2][3]
• For every integer ${\displaystyle n\geq 2}$, the free factor graph ${\displaystyle {\mathcal {F}}_{n}^{(1)}}$, equipped with the simplicial metric, is Gromov-hyperbolic. This result was originally established by Bestvina and Feighn;[4] see also [5][6] for subsequent alternative proofs.
• An element ${\displaystyle \varphi \in \operatorname {Out} (F_{n})}$ acts as a loxodromic isometry of ${\displaystyle {\mathcal {F}}_{n}^{(1)}}$ if and only if ${\displaystyle \varphi }$ is fully irreducible.[4]
• There exists a coarsely Lipschitz coarsely ${\displaystyle \operatorname {Out} (F_{n})}$-equivariant coarsely surjective map ${\displaystyle {\mathcal {FS}}_{n}\to {\mathcal {F}}_{n}^{(1)}}$, where ${\displaystyle {\mathcal {FS}}_{n}}$ 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. [7]
• Similarly, there exists a natural coarsely Lipschitz coarsely ${\displaystyle \operatorname {Out} (F_{n})}$-equivariant coarsely surjective map ${\displaystyle CV_{n}\to {\mathcal {F}}_{n}^{(1)}}$, where ${\displaystyle CV_{n}}$ is the (volume-ones normalized) Culler–Vogtmann Outer space, equipped with the symmetric Lipschitz metric. The map ${\displaystyle \pi }$ takes a geodesic path in ${\displaystyle CV_{n}}$ to a path in ${\displaystyle {\mathcal {F}}F_{n}}$ contained in a uniform Hausdorff neighborhood of the geodesic with the same endpoints.[4]
• The hyperbolic boundary ${\displaystyle \partial {\mathcal {F}}_{n}^{(1)}}$ of the free factor graph can be identified with the set of equivalence classes of "arational" ${\displaystyle F_{n}}$-trees in the boundary ${\displaystyle \partial CV_{n}}$ of the Outer space ${\displaystyle CV_{n}}$.[8]
• The free factor complex is a key tool in studying the behavior of random walks on ${\displaystyle \operatorname {Out} (F_{n})}$ and in identifying the Poisson boundary of ${\displaystyle \operatorname {Out} (F_{n})}$.[9]

Other models

There are several other models which produce graphs coarsely ${\displaystyle \operatorname {Out} (F_{n})}$-equivariantly quasi-isometric to ${\displaystyle {\mathcal {F}}_{n}^{(1)}}$. These models include:

• The graph whose vertex set is ${\displaystyle {\mathcal {F}}_{n}^{0}}$ and where two distinct vertices ${\displaystyle v_{0},v_{1}}$ are adjacent if and only if there exists a free product decomposition ${\displaystyle F_{n}=A\ast B\ast C}$ such that ${\displaystyle v_{0}=[A]}$ and ${\displaystyle v_{1}=[B]}$.
• The free bases graph whose vertex set is the set of ${\displaystyle F_{n}}$-conjugacy classes of free bases of ${\displaystyle F_{n}}$, and where two vertices ${\displaystyle v_{0},v_{1}}$ are adjacent if and only if there exist free bases ${\displaystyle {\mathcal {A}},{\mathcal {B}}}$ of ${\displaystyle F_{n}}$ such that ${\displaystyle v_{0}=[{\mathcal {A}}],v_{1}=[{\mathcal {B}}]}$ and ${\displaystyle {\mathcal {A}}\cap {\mathcal {B}}\neq \varnothing }$.[5]

References

1. Allen Hatcher, and Karen Vogtmann, The complex of free factors of a free group. Quarterly Journal of Mathematics, Oxford Ser. (2) 49 (1998), no. 196, pp. 459–468
2. Ilya Kapovich and Martin Lustig, Geometric intersection number and analogues of the curve complex for free groups. Geometry & Topology 13 (2009), no. 3, pp. 1805–1833
3. Jason Behrstock, Mladen Bestvina, and Matt Clay, Growth of intersection numbers for free group automorphisms. Journal of Topology 3 (2010), no. 2, pp. 280–310
4. Mladen Bestvina and Mark Feighn, Hyperbolicity of the complex of free factors. Advances in Mathematics 256 (2014), pp. 104–155
5. Ilya Kapovich and Kasra Rafi, On hyperbolicity of free splitting and free factor complexes. Groups, Geometry, and Dynamics 8 (2014), no. 2, pp. 391–414
6. Arnaud Hilion and Camille Horbez, The hyperbolicity of the sphere complex via surgery paths, Journal für die reine und angewandte Mathematik 730 (2017), 135–161
7. Michael Handel and Lee Mosher, The free splitting complex of a free group, I: hyperbolicity. Geometry & Topology, 17 (2013), no. 3, 1581--1672. MR3073931doi:10.2140/gt.2013.17.1581
8. Mladen Bestvina and Patrick Reynolds, The boundary of the complex of free factors. Duke Mathematical Journal 164 (2015), no. 11, pp. 2213–2251
9. Camille Horbez, The Poisson boundary of ${\displaystyle \operatorname {Out} (F_{N})}$. Duke Mathematical Journal 165 (2016), no. 2, pp. 341–369

See also

• Mapping class group