Free factor complex explained

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{Out}(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 \{ 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 of rank n . The free factor complex l 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 l F ^{( 0 )} _{n} = \{[A] | A\le F _{n} isaproperfreefactorof F _{n} \}.

(2) For

k\ge 1 , a k -simplex in l F _{n} is a collection of k +1 distinct 0-cells \{v _{0,} v _{1,} ... , v _{k\}\subset} l F ^{( 0 )} _{n} such that there exist free factors A _{0,A} _{1,...,} A _{k} of F _{n} such that v _{i=A} _{i} for i =0 , 1 , ... , k , and that A _{0\le} A _{1\le} ... \le A _{k} . [The assumption that these 0-cells are distinct implies that <math>A_i\ne A_{i+1}</math> for <math> i=0,1,\dots, k-1</math>]. 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 ln eq B .

For

n =2 the above definition produces a complex with no k -cells of dimension k\ge 1 . Therefore, l F _{2} is defined slightly differently. One still defines l F ^{( 0 )} _{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 \{v _{0,v} _{1\}\subset} l F ^{( 0 )} _{2} determine a 1-simplex in l 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 l F _{2} has no k -cells of dimension k\ge 2 .

For

n\ge 2 the 1-skeleton l F ^{( 1 )} _{n} is called the free factor graph for F _{n} .

Main properties

• For every integer

n\ge 3 the complex l F _{n} is connected, locally infinite, and has dimension n -2 . The complex l F _{2} is connected, locally infinite, and has dimension 1.

• For

n =2 , the graph l F _{2} is isomorphic to the Farey graph.

• There is a natural action of \operatorname{Out}(F _{n)} on

l F _{n} by simplicial automorphisms. For a k-simplex \Delta = \{[A _{0],...,} [A _{k]\}} and \varphi\in \operatorname{Out}(F _{n)} one has \varphi \Delta: = \{[\varphi(A _{0)],...,} [\varphi(A _{k)]\}} .

• For

n\ge 3 the complex l F _{n} has the homotopy type of a wedge of spheres of dimension n -2 .^[1]

• For every integer

n\ge 2 , the free factor graph l F ^{( 1 )} _{n} , 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 l F ^{( 1 )} _{n} , 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

\varphi\in \operatorname{Out}(F _{n)} acts as a loxodromic isometry of l F ^{( 1 )} _{n} if and only if \varphi is fully irreducible.^[4]

• There exists a coarsely Lipschitz coarsely

\operatorname{Out}(F _{n)} -equivariant coarsely surjective map l{FS} _{n\to} l F ^{( 1 )} _{n} , where l{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

\operatorname{Out}(F _{n)} -equivariant coarsely surjective map CV _{n\to} l F ^{( 1 )} _{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 l FF _{n} contained in a uniform Hausdorff neighborhood of the geodesic with the same endpoints.^[4]

• The hyperbolic boundary

\partial l F ^{( 1 )} _{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} .^[8]

• The free factor complex is a key tool in studying the behavior of random walks on

\operatorname{Out}(F _{n)} and in identifying the Poisson boundary of \operatorname{Out}(F _{n)} .^[9]

Other models

There are several other models which produce graphs coarsely

\operatorname{Out}(F _{n)} -equivariantly quasi-isometric to

l F ^{( 1 )} _{n} . These models include:

• The graph whose vertex set is

l F ^{0} _{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=[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 l A, l B of F _{n} such that v _{0=[lA],} v _{1=[lB]} and l A\cap l B\ne \varnothing .^[5]

See also

• Mapping class group

Notes and References

1. Hatcher . Allen . Allen Hatcher. Vogtmann . Karen . Karen Vogtmann. The complex of free factors of a free group. . Series 2. 49. 1998. 196. 459–468. 10.1093/qmathj/49.4.459. 2203.15602.
2. Kapovich . Ilya . Ilya Kapovich. Lustig . Martin. Geometric intersection number and analogues of the curve complex for free groups. Geometry & Topology. 13. 2009. 3. 1805–1833. 10.2140/gt.2009.13.1805 . free. 0711.3806.
3. Behrstock . Jason. Bestvina . Mladen . Mladen Bestvina. Clay . Matt. Growth of intersection numbers for free group automorphisms. Journal of Topology. 3. 2010. 2. 280–310. 10.1112/jtopol/jtq008. 0806.4975.
4. Bestvina . Mladen . Mladen Bestvina. Feighn . Mark. Hyperbolicity of the complex of free factors. Advances in Mathematics. 256. 2014. 104–155. 10.1016/j.aim.2014.02.001 . free. 1107.3308.
5. Kapovich . Ilya . Ilya Kapovich. Rafi . Kasra. On hyperbolicity of free splitting and free factor complexes. Groups, Geometry, and Dynamics. 8. 2014. 2. 391–414. 10.4171/GGD/231 . free. 1206.3626.
6. Hilion . Arnaud. Horbez . Camille. The hyperbolicity of the sphere complex via surgery paths. Journal für die reine und angewandte Mathematik. 730. 2017. 135–161. 10.1515/crelle-2014-0128. 1210.6183.
7. Handel . Michael . Michael Handel. Mosher . Lee. The free splitting complex of a free group, I: hyperbolicity. Geometry & Topology. 17. 2013. 3. 1581–1672. 3073931. 10.2140/gt.2013.17.1581 . free. 1111.1994.
8. Bestvina . Mladen . Mladen Bestvina. Reynolds . Patrick. The boundary of the complex of free factors. Duke Mathematical Journal. 164. 2015. 11. 2213–2251. 10.1215/00127094-3129702. 1211.3608.
9. Horbez . Camille. The Poisson boundary of \operatorname{Out}(F _{N)} . Duke Mathematical Journal. 165. 2016. 2. 341–369. 10.1215/00127094-3166308. 1405.7938.