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

. The free factor complex plays a significant role in the study of large-scale geometry of ${\displaystyle \operatorname {Out} (F_{n})}$.

Formal definition

For a free group ${\displaystyle G}$ a proper free factor of ${\displaystyle G}$ is a subgroup ${\displaystyle A\leq G}$ such that ${\displaystyle A\neq \{1\},A\neq G}$ and that there exists a subgroup ${\displaystyle B\leq G}$ such that ${\displaystyle G=A\ast B}$.

Let ${\displaystyle n\geq 3}$ be an integer and let ${\displaystyle F_{n}}$ be the free group of rank ${\displaystyle n}$. The free factor complex ${\displaystyle {\mathcal {F}}_{n}}$ for ${\displaystyle F_{n}}$ is a simplicial complex where:

(1) The 0-cells are the conjugacy classes in ${\displaystyle F_{n}}$ of proper free factors of ${\displaystyle F_{n}}$, that is

${\displaystyle {\mathcal {F}}_{n}^{(0)}=\{[A]|A\leq F_{n}{\text{ is a proper free factor of }}F_{n}\}.}$

(2) For ${\displaystyle k\geq 1}$, a ${\displaystyle k}$-simplex in ${\displaystyle {\mathcal {F}}_{n}}$ is a collection of ${\displaystyle k+1}$ distinct 0-cells ${\displaystyle \{v_{0},v_{1},\dots ,v_{k}\}\subset {\mathcal {F}}_{n}^{(0)}}$ such that there exist free factors ${\displaystyle A_{0},A_{1},\dots ,A_{k}}$ of ${\displaystyle F_{n}}$ such that ${\displaystyle v_{i}=A_{i}}$ for ${\displaystyle i=0,1,\dots ,k}$, and that ${\displaystyle A_{0}\leq A_{1}\leq \dots \leq A_{k}}$. [The assumption that these 0-cells are distinct implies that ${\displaystyle A_{i}\neq A_{i+1}}$ for ${\displaystyle i=0,1,\dots ,k-1}$]. In particular, a 1-cell is a collection ${\displaystyle \{[A],[B]\}}$ of two distinct 0-cells where ${\displaystyle A,B\leq F_{n}}$ are proper free factors of ${\displaystyle F_{n}}$ such that ${\displaystyle A\lneq B}$.

For ${\displaystyle n=2}$ the above definition produces a complex with no ${\displaystyle k}$-cells of dimension ${\displaystyle k\geq 1}$. Therefore, ${\displaystyle {\mathcal {F}}_{2}}$ is defined slightly differently. One still defines ${\displaystyle {\mathcal {F}}_{2}^{(0)}}$ to be the set of conjugacy classes of proper free factors of ${\displaystyle F_{2}}$; (such free factors are necessarily infinite cyclic). Two distinct 0-simplices ${\displaystyle \{v_{0},v_{1}\}\subset {\mathcal {F}}_{2}^{(0)}}$ determine a 1-simplex in ${\displaystyle {\mathcal {F}}_{2}}$ if and only if there exists a free basis ${\displaystyle a,b}$ of ${\displaystyle F_{2}}$ such that ${\displaystyle v_{0}=[\langle a\rangle ],v_{1}=[\langle b\rangle ]}$. The complex ${\displaystyle {\mathcal {F}}_{2}}$ has no ${\displaystyle k}$-cells of dimension ${\displaystyle k\geq 2}$.

For ${\displaystyle n\geq 2}$ the 1-skeleton ${\displaystyle {\mathcal {F}}_{n}^{(1)}}$ is called the free factor graph for ${\displaystyle F_{n}}$.

Main properties

• For every integer ${\displaystyle n\geq 3}$ the complex ${\displaystyle {\mathcal {F}}_{n}}$ is connected, locally infinite, and has dimension ${\displaystyle n-2}$. The complex ${\displaystyle {\mathcal {F}}_{2}}$ is connected, locally infinite, and has dimension 1.
• For ${\displaystyle n=2}$, the graph ${\displaystyle {\mathcal {F}}_{2}}$ is isomorphic to the
  Farey graph
  .
• There is a natural
  action
  of ${\displaystyle \operatorname {Out} (F_{n})}$ on ${\displaystyle {\mathcal {F}}_{n}}$ by simplicial automorphisms. For a k-simplex ${\displaystyle \Delta =\{[A_{0}],\dots ,[A_{k}]\}}$ and ${\displaystyle \varphi \in \operatorname {Out} (F_{n})}$ one has ${\displaystyle \varphi \Delta :=\{[\varphi (A_{0})],\dots ,[\varphi (A_{k})]\}}$.
• For ${\displaystyle n\geq 3}$ the complex ${\displaystyle {\mathcal {F}}_{n}}$ has the
  homotopy type
  of a wedge of spheres of dimension ${\displaystyle n-2}$.^[1]
• For every integer ${\displaystyle n\geq 2}$, the free factor graph ${\displaystyle {\mathcal {F}}_{n}^{(1)}}$, equipped with the simplicial metric (where every edge has length 1), is a connected graph of infinite diameter.^[2][3]
• For every integer ${\displaystyle n\geq 2}$, the free factor graph ${\displaystyle {\mathcal {F}}_{n}^{(1)}}$, 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 ${\displaystyle \varphi \in \operatorname {Out} (F_{n})}$ acts as a loxodromic isometry of ${\displaystyle {\mathcal {F}}_{n}^{(1)}}$ if and only if ${\displaystyle \varphi }$ is fully irreducible.^[4]
• There exists a coarsely Lipschitz coarsely ${\displaystyle \operatorname {Out} (F_{n})}$-equivariant coarsely surjective map ${\displaystyle {\mathcal {FS}}_{n}\to {\mathcal {F}}_{n}^{(1)}}$, where ${\displaystyle {\mathcal {FS}}_{n}}$ is the
  Gromov-hyperbolic, as was proved by Handel and Mosher.^[7]
• Similarly, there exists a natural coarsely Lipschitz coarsely ${\displaystyle \operatorname {Out} (F_{n})}$-equivariant coarsely surjective map ${\displaystyle CV_{n}\to {\mathcal {F}}_{n}^{(1)}}$, where ${\displaystyle CV_{n}}$ is the (volume-ones normalized) Culler–Vogtmann Outer space, equipped with the symmetric Lipschitz metric. The map ${\displaystyle \pi }$ takes a geodesic path in ${\displaystyle CV_{n}}$ to a path in ${\displaystyle {\mathcal {F}}F_{n}}$ contained in a uniform Hausdorff neighborhood of the geodesic with the same endpoints.^[4]
• The hyperbolic boundary ${\displaystyle \partial {\mathcal {F}}_{n}^{(1)}}$ of the free factor graph can be identified with the set of equivalence classes of "arational" ${\displaystyle F_{n}}$-trees in the boundary ${\displaystyle \partial CV_{n}}$ of the Outer space ${\displaystyle CV_{n}}$.^[8]
• The free factor complex is a key tool in studying the behavior of random walks on ${\displaystyle \operatorname {Out} (F_{n})}$ and in identifying the Poisson boundary of ${\displaystyle \operatorname {Out} (F_{n})}$.^[9]

Other models

There are several other models which produce graphs coarsely ${\displaystyle \operatorname {Out} (F_{n})}$-equivariantly quasi-isometric to ${\displaystyle {\mathcal {F}}_{n}^{(1)}}$. These models include:

• The graph whose vertex set is ${\displaystyle {\mathcal {F}}_{n}^{0}}$ and where two distinct vertices ${\displaystyle v_{0},v_{1}}$ are adjacent if and only if there exists a free product decomposition ${\displaystyle F_{n}=A\ast B\ast C}$ such that ${\displaystyle v_{0}=[A]}$ and ${\displaystyle v_{1}=[B]}$.
• The free bases graph whose vertex set is the set of ${\displaystyle F_{n}}$-conjugacy classes of free bases of ${\displaystyle F_{n}}$, and where two vertices ${\displaystyle v_{0},v_{1}}$ are adjacent if and only if there exist free bases ${\displaystyle {\mathcal {A}},{\mathcal {B}}}$ of ${\displaystyle F_{n}}$ such that ${\displaystyle v_{0}=[{\mathcal {A}}],v_{1}=[{\mathcal {B}}]}$ and ${\displaystyle {\mathcal {A}}\cap {\mathcal {B}}\neq \varnothing }$.^[5]

References

See also

• Mapping class group