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
For a free group
a proper free factor of
and that there exists a subgroup
be an integer and let
be the free group of rank
The free factor complex
is a simplicial complex where: (1) The 0-cells are the conjugacy classes in
of proper free factors of
is a collection of
distinct 0-cells
such that there exist free factors
[The assumption that these 0-cells are distinct implies that
In particular, a 1-cell is a collection
of two distinct 0-cells where
are proper free factors of
the above definition produces a complex with no
-cells of dimension
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).
if and only if there exists a free basis
-cells of dimension
is called the free factor graph for
There are several other models which produce graphs coarsely
These models include: