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. 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 formula_1.
Formal definition.
For a free group formula_2 a "proper free factor" of formula_2 is a subgroup formula_4 such that formula_5 and that there exists a subgroup formula_6 such that formula_7.
Let formula_8 be an integer and let formula_9 be the free group of rank formula_10. The free factor complex formula_11 for formula_9 is a simplicial complex where:
(1) The 0-cells are the conjugacy classes in formula_9 of proper free factors of formula_9, that is 
(2) For formula_16, a formula_17-simplex in formula_11 is a collection of formula_19 distinct 0-cells formula_20 such that there exist free factors formula_21 of formula_9 such that formula_23 for formula_24, and that formula_25. [The assumption that these 0-cells are distinct implies that formula_26 for formula_27]. In particular, a 1-cell is a collection formula_28 of two distinct 0-cells where formula_29 are proper free factors of formula_9 such that formula_31.
For formula_32 the above definition produces a complex with no formula_17-cells of dimension formula_16. Therefore, formula_35 is defined slightly differently. One still defines formula_36 to be the set of conjugacy classes of proper free factors of formula_37; (such free factors are necessarily infinite cyclic). Two distinct 0-simplices formula_38 determine a 1-simplex in formula_35 if and only if there exists a free basis formula_40 of formula_37 such that formula_42.
The complex formula_35 has no formula_17-cells of dimension formula_45.
For formula_46 the 1-skeleton formula_47 is called the free factor graph for formula_9.
Other models.
There are several other models which produce graphs coarsely formula_1-equivariantly quasi-isometric to formula_47. These models include: