free factor complex

For a free group $G$ a proper free factor of $G$ is a subgroup $A\backslash le\; G$ such that $A\backslash ne\; \backslash \{1\backslash \},\; A\backslash ne\; G$ and that there exists a subgroup $B\backslash le\; G$ such that $G=A\backslash ast\; B$.

Let $n\backslash ge\; 3$ be an integer and let $F\_n$ be the free group of rank $n$. The free factor complex $\backslash mathcal\; F\_n$ 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

:$\backslash mathcal\; F\_n^\{(0)\}=\backslash \{[A]\; |\; A\backslash le\; F\_n\; \backslash text\{\; is\; a\; proper\; free\; factor\; of\; \}\; F\_n\; \backslash \}.$

(2) For $k\backslash ge\; 1$, a $k$-simplex in $\backslash mathcal\; F\_n$ is a collection of $k+1$ distinct 0-cells $\backslash \{v\_0,\; v\_1,\; \backslash dots,\; v\_k\backslash \}\backslash subset\; \backslash mathcal\; F\_n^\{(0)\}$ such that there exist free factors $A\_0,A\_1,\backslash dots,\; A\_k$ of $F\_n$ such that $v\_i=A\_i$ for $i=0,1,\backslash dots,\; k$, and that $A\_0\backslash le\; A\_1\backslash le\; \backslash dots\; \backslash le\; A\_k$. [The assumption that these 0-cells are distinct implies that $A\_i\backslash ne\; A\_\{i+1\}$ for $i=0,1,\backslash dots,\; k-1$]. In particular, a 1-cell is a collection $\backslash \{[A],\; [B]\backslash \}$ of two distinct 0-cells where $A,B\backslash le\; F\_n$ are proper free factors of $F\_n$ such that $A\backslash lneq\; B$.

For $n=2$ the above definition produces a complex with no $k$-cells of dimension $k\backslash ge\; 1$. Therefore, $\backslash mathcal\; F\_2$ is defined slightly differently. One still defines $\backslash mathcal\; F\_2^\{(0)\}$ 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 $\backslash \{v\_0,v\_1\backslash \}\backslash subset\; \backslash mathcal\; F\_2^\{(0)\}$ determine a 1-simplex in $\backslash mathcal\; F\_2$ if and only if there exists a free basis $a,b$ of $F\_2$ such that $v\_0=[\backslash langle\; a\backslash rangle],\; v\_1=[\backslash langle\; b\backslash rangle]$.

The complex $\backslash mathcal\; F\_2$ has no $k$-cells of dimension $k\backslash ge\; 2$.

For $n\backslash ge\; 2$ the 1-skeleton $\backslash mathcal\; F\_n^\{(1)\}$ is called the free factor graph for $F\_n$.

• For every integer $n\backslash ge\; 3$ the complex $\backslash mathcal\; F\_n$ is connected, locally infinite, and has dimension $n-2$. The complex $\backslash mathcal\; F\_2$ is connected, locally infinite, and has dimension 1.
• For $n=2$, the graph $\backslash mathcal\; F\_2$ is isomorphic to the Farey graph.
• There is a natural action of $\backslash operatorname\{Out\}(F\_n)$ on $\backslash mathcal\; F\_n$ by simplicial automorphisms. For a k-simplex $\backslash Delta=\backslash \{[A\_0],\backslash dots,\; [A\_k]\backslash \}$ and $\backslash varphi\backslash in\; \backslash operatorname\{Out\}(F\_n)$ one has $\backslash varphi\; \backslash Delta:=\backslash \{[\backslash varphi(A\_0)],\backslash dots,\; [\backslash varphi(A\_k)]\backslash \}$.
• For $n\backslash ge\; 3$ the complex $\backslash mathcal\; F\_n$ has the homotopy type of a wedge of spheres of dimension $n-2$.{{cite journal

| last1=Hatcher | first1=Allen | authorlink1=Allen Hatcher

| last2=Vogtmann | first2=Karen | authorlink2=Karen Vogtmann

| title=The complex of free factors of a free group

| journal=Quarterly Journal of Mathematics | series=Series 2

| volume=49

| date=1998

| issue=196

| pages=459–468

| doi=10.1093/qmathj/49.4.459| arxiv=2203.15602

}}

• For every integer $n\backslash ge\; 2$, the free factor graph $\backslash mathcal\; F\_n^\{(1)\}$, equipped with the simplicial metric (where every edge has length 1), is a connected graph of infinite diameter.{{cite journal

| last1=Kapovich | first1=Ilya | authorlink1=Ilya Kapovich

| last2=Lustig | first2=Martin

| title=Geometric intersection number and analogues of the curve complex for free groups

| journal=Geometry & Topology

| volume=13

| date=2009

| issue=3

| pages=1805–1833

| doi=10.2140/gt.2009.13.1805 | doi-access=free| arxiv=0711.3806

}}{{cite journal

| last1=Behrstock | first1=Jason

| last2=Bestvina | first2=Mladen | authorlink2=Mladen Bestvina

| last3=Clay | first3=Matt

| title=Growth of intersection numbers for free group automorphisms

| journal=Journal of Topology

| volume=3

| date=2010

| issue=2

| pages=280–310

| doi=10.1112/jtopol/jtq008| arxiv=0806.4975

}}

• For every integer $n\backslash ge\; 2$, the free factor graph $\backslash mathcal\; F\_n^\{(1)\}$, equipped with the simplicial metric, is Gromov-hyperbolic. This result was originally established by Mladen Bestvina and Mark Feighn;{{cite journal

| last1=Bestvina | first1=Mladen | authorlink1=Mladen Bestvina

| last2=Feighn | first2=Mark

| title=Hyperbolicity of the complex of free factors

| journal=Advances in Mathematics

| volume=256

| date=2014

| pages=104–155

| doi=10.1016/j.aim.2014.02.001 | doi-access=free| arxiv=1107.3308

}} see also {{cite journal

| last1=Kapovich | first1=Ilya | authorlink1=Ilya Kapovich

| last2=Rafi | first2=Kasra

| title=On hyperbolicity of free splitting and free factor complexes

| journal=Groups, Geometry, and Dynamics

| volume=8

| date=2014

| issue=2

| pages=391–414

| doi=10.4171/GGD/231 | doi-access=free| arxiv=1206.3626

}}{{cite journal

| last1=Hilion | first1=Arnaud

| last2=Horbez | first2=Camille

| title=The hyperbolicity of the sphere complex via surgery paths

| journal=Journal für die reine und angewandte Mathematik

| volume=730

| date=2017

| pages=135–161

| doi=10.1515/crelle-2014-0128| arxiv=1210.6183

}} for subsequent alternative proofs.

• An element $\backslash varphi\backslash in\; \backslash operatorname\{Out\}(F\_n)$ acts as a loxodromic isometry of $\backslash mathcal\; F\_n^\{(1)\}$ if and only if $\backslash varphi$ is fully irreducible.
• There exists a coarsely Lipschitz coarsely $\backslash operatorname\{Out\}(F\_n)$-equivariant coarsely surjective map $\backslash mathcal\{FS\}\_n\backslash to\; \backslash mathcal\; F\_n^\{(1)\}$, where $\backslash mathcal\{FS\}\_n$ 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.{{cite journal

| last1=Handel | first1=Michael | authorlink1=Michael Handel

| last2=Mosher | first2=Lee

| title=The free splitting complex of a free group, I: hyperbolicity

| journal=Geometry & Topology

| volume=17

| date=2013

| issue=3

| pages=1581–1672

| mr=3073931

| doi=10.2140/gt.2013.17.1581 | doi-access=free| arxiv=1111.1994

}}

• Similarly, there exists a natural coarsely Lipschitz coarsely $\backslash operatorname\{Out\}(F\_n)$-equivariant coarsely surjective map $CV\_n\backslash to\; \backslash mathcal\; F\_n^\{(1)\}$, where $CV\_n$ is the (volume-ones normalized) Culler–Vogtmann Outer space, equipped with the symmetric Lipschitz metric. The map $\backslash pi$ takes a geodesic path in $CV\_n$ to a path in $\backslash mathcal\; FF\_n$ contained in a uniform Hausdorff neighborhood of the geodesic with the same endpoints.
• The hyperbolic boundary $\backslash partial\; \backslash mathcal\; F\_n^\{(1)\}$ of the free factor graph can be identified with the set of equivalence classes of "arational" $F\_n$-trees in the boundary $\backslash partial\; CV\_n$ of the Outer space $CV\_n$.{{cite journal

| last1=Bestvina | first1=Mladen | authorlink1=Mladen Bestvina

| last2=Reynolds | first2=Patrick

| title=The boundary of the complex of free factors

| journal=Duke Mathematical Journal

| volume=164

| date=2015

| issue=11

| pages=2213–2251

| doi=10.1215/00127094-3129702| arxiv=1211.3608

}}

• The free factor complex is a key tool in studying the behavior of random walks on $\backslash operatorname\{Out\}(F\_n)$ and in identifying the Poisson boundary of $\backslash operatorname\{Out\}(F\_n)$.{{cite journal

| last1=Horbez | first1=Camille

| title=The Poisson boundary of $\backslash operatorname\{Out\}(F\_N)$

| journal=Duke Mathematical Journal

| volume=165

| date=2016

| issue=2

| pages=341–369

| doi=10.1215/00127094-3166308| arxiv=1405.7938

}}

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

• The graph whose vertex set is $\backslash mathcal\; F\_n^\{0\}$ and where two distinct vertices $v\_0,v\_1$ are adjacent if and only if there exists a free product decomposition $F\_n=A\backslash ast\; B\backslash ast\; C$ 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 $\backslash mathcal\; A,\; \backslash mathcal\; B$ of $F\_n$ such that $v\_0=[\backslash mathcal\; A],\; v\_1=[\backslash mathcal\; B]$ and $\backslash mathcal\; A\backslash cap\; \backslash mathcal\; B\backslash ne\; \backslash varnothing$.

{{Reflist}}

• Mapping class group

Category:Geometric group theory

Category:Geometric topology