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 \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 \mathcal 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

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

See also