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 .

Formal definition

For a free group a proper free factor of is a subgroup such that and that there exists a subgroup such that .

Let be an integer and let be the free group of rank . The free factor complex for is a simplicial complex where:

(1) The 0-cells are the conjugacy classes in of proper free factors of , that is

(2) For , a -simplex in is a collection of distinct 0-cells such that there exist free factors of such that for , and that . [The assumption that these 0-cells are distinct implies that for ]. In particular, a 1-cell is a collection of two distinct 0-cells where are proper free factors of such that .

For the above definition produces a complex with no -cells of dimension . Therefore, is defined slightly differently. One still defines to be the set of conjugacy classes of proper free factors of ; (such free factors are necessarily infinite cyclic). Two distinct 0-simplices determine a 1-simplex in if and only if there exists a free basis of such that . The complex has no -cells of dimension .

For the 1-skeleton is called the free factor graph for .

Main properties

  • For every integer the complex is connected, locally infinite, and has dimension . The complex is connected, locally infinite, and has dimension 1.
  • For , the graph is isomorphic to the Farey graph.
  • There is a natural action of on by simplicial automorphisms. For a k-simplex and one has .
  • For the complex has the homotopy type of a wedge of spheres of dimension .[1]
  • For every integer , the free factor graph , equipped with the simplicial metric (where every edge has length 1), is a connected graph of infinite diameter.[2][3]
  • For every integer , the free factor graph , 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 acts as a loxodromic isometry of if and only if is fully irreducible.[4]
  • There exists a coarsely Lipschitz coarsely -equivariant coarsely surjective map , where 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 -equivariant coarsely surjective map , where is the (volume-ones normalized) Culler–Vogtmann Outer space, equipped with the symmetric Lipschitz metric. The map takes a geodesic path in to a path in contained in a uniform Hausdorff neighborhood of the geodesic with the same endpoints.[4]
  • The hyperbolic boundary of the free factor graph can be identified with the set of equivalence classes of "arational" -trees in the boundary of the Outer space .[8]
  • The free factor complex is a key tool in studying the behavior of random walks on and in identifying the Poisson boundary of .[9]

Other models

There are several other models which produce graphs coarsely -equivariantly quasi-isometric to . These models include:

  • The graph whose vertex set is and where two distinct vertices are adjacent if and only if there exists a free product decomposition such that and .
  • The free bases graph whose vertex set is the set of -conjugacy classes of free bases of , and where two vertices are adjacent if and only if there exist free bases of such that and .[5]

References

  1. 1 2 Hatcher, Allen; Vogtmann, Karen (1998). "The complex of free factors of a free group". Quarterly Journal of Mathematics. Series 2. 49 (196): 459–468. 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. 1 2 3 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. 1 2 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 ". Duke Mathematical Journal. 165 (2): 341–369. arXiv:1405.7938. doi:10.1215/00127094-3166308.

See also

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