The boundary of the complex of free factors.

*(English)*Zbl 1337.20040Let \(F_N\) be the free group of free rank \(N\). With \(F_N\) is associated a complex \(\mathcal F=\mathcal F_N\) of free factors. Vertices of \(\mathcal F_N\) are conjugacy classes of non trivial proper free factors of \(F_N\) and higher-dimensional simplices correspond to chains of inclusions of free factors. \(\mathcal F_N\) is equipped with the simplicial metric and in [M. Bestvina and M. Feighn, Adv. Math. 256, 104-155 (2014; Zbl 1348.20028); corrigendum 259, 843 (2014)] it is proved that \(\mathcal F_N\) is Gromov hyperbolic. In the present paper the authors give a concrete description of the boundary \(\partial\mathcal F_N\) of \(\mathcal F_N\).

Before stating the main result we quote the necessary definitions and terminology. The unprojectivized outer space of rank \(N\), denoted \(cv_N\), is the topological space whose underlying set consists of free, minimal, discrete, isometric actions of \(F_N\) on \(\mathbb R\)-trees. A minimal \(F_N\)-tree is completely determined by its translation length function [M. Culler and J. W. Morgan, Proc. Lond. Math. Soc., III. Ser. 55, 571-604 (1987; Zbl 0658.20021)]. This gives an inclusion \(cv_N\subseteq\mathbb R^{F_N}\) and a topology on \(cv_N\). Let \(\partial cv_N=\overline{cv_N}\setminus cv_N\) denote the boundary of \(cv_N\), where \(\overline{cv_N}\) is the closure of \(cv_N\) in \(\mathbb R^{F_N}\).

The image of \(cv_N\) in the projective space \(\mathbb{PR}^{F_N}\) is the Culler-Vogtmann Outer space \(CV_N\) and the boundary \(\partial CV_N=\overline{CV_N}\setminus CV_{N}\) is the image of \(\partial cv_N=\overline{cv_N}\setminus cv_N\).

Let \(\partial F_N\) be the Gromov boundary of \(F_N\), that is the boundary of any Cayley graph of \(F_N\). Let \(\partial^2(F_N)=\partial F_N\times\partial F_N\setminus\Delta\), where \(\Delta\) is the diagonal. The left action of \(F_N\) on a Cayley graph induces actions by homeomorphisms of \(F_N\) on \(\partial F_N\) and on \(\partial^2(F_N)\). Let \(i\colon\partial^2(F_N)\to\partial^2(F_N)\) denote the involution (the flip) that exchanges the factors. A lamination is a nonempty, closed \(F_N\)-invariant, \(i\)-invariant subset \(L\subseteq\partial^2(F_N)\). The elements of a lamination \(X\) are called leaves and it is called arational if no leaf is carried by a proper free factor of \(F_N\).

Associated to a \(T\in\partial cv_N\) is a lamination \(L(T)\), which is constructed as follows. Let \(L_\varepsilon(T)=\overline{\{(g^{-\infty},g^\infty\mid\ell_T(g)<\varepsilon\}}\), and define \(L(T)=\bigcap_{\varepsilon>0}L_\varepsilon(T)\). Here, for a \(1\neq g\in F_N\), \(g^{-\infty}\) and \(g^\infty\) denote the attracting and repelling fixed points of \(g\) in \(\partial F_N\) and \(\ell_T\) is a length function on \(T\).

A tree \(T\in\partial CV_N\) is called arational if the lamination \(L(T)\) is arational. Let \(\mathcal{AT}\subseteq\partial CV_N\) denote the set of arational trees, equiped with the subspace topology. Define a relation \(\sim\) on \(\mathcal{AT}\) by \(S\sim T\) if and only if \(L(S)=L(T)\), and give \(\mathcal{AT}/\sim\) the quotient topology.

Now we can state the main theorem. Theorem. The space \(\partial\mathcal F\) is homeomorphic to \(\mathcal{AT}/\sim\).

The arguments of the authors use the geometry of Outer space and folding paths as developed by M. Bestvina and M. Feighn [op. cit.] and the structure theory of trees in \(\partial CV_N\) developed by Coulbois, Hilion, Lustig, and Reynolds in recent series of papers.

As the authors point out the theorem above is a very strong analogy of E. Klarreich’s description, [in The boundary at infinity of the complex of curves and the relative Teichmüller space, preprint https://pressfolios-production.s3.amazonaws.com/uploads/story/story-pdf/145710/1457101434403642.pdf], of the boundary \(\partial\mathcal C(S)\) of the complex of curves \(\mathcal C(S)\) associated to a nonexceptional surface \(S\).

Before stating the main result we quote the necessary definitions and terminology. The unprojectivized outer space of rank \(N\), denoted \(cv_N\), is the topological space whose underlying set consists of free, minimal, discrete, isometric actions of \(F_N\) on \(\mathbb R\)-trees. A minimal \(F_N\)-tree is completely determined by its translation length function [M. Culler and J. W. Morgan, Proc. Lond. Math. Soc., III. Ser. 55, 571-604 (1987; Zbl 0658.20021)]. This gives an inclusion \(cv_N\subseteq\mathbb R^{F_N}\) and a topology on \(cv_N\). Let \(\partial cv_N=\overline{cv_N}\setminus cv_N\) denote the boundary of \(cv_N\), where \(\overline{cv_N}\) is the closure of \(cv_N\) in \(\mathbb R^{F_N}\).

The image of \(cv_N\) in the projective space \(\mathbb{PR}^{F_N}\) is the Culler-Vogtmann Outer space \(CV_N\) and the boundary \(\partial CV_N=\overline{CV_N}\setminus CV_{N}\) is the image of \(\partial cv_N=\overline{cv_N}\setminus cv_N\).

Let \(\partial F_N\) be the Gromov boundary of \(F_N\), that is the boundary of any Cayley graph of \(F_N\). Let \(\partial^2(F_N)=\partial F_N\times\partial F_N\setminus\Delta\), where \(\Delta\) is the diagonal. The left action of \(F_N\) on a Cayley graph induces actions by homeomorphisms of \(F_N\) on \(\partial F_N\) and on \(\partial^2(F_N)\). Let \(i\colon\partial^2(F_N)\to\partial^2(F_N)\) denote the involution (the flip) that exchanges the factors. A lamination is a nonempty, closed \(F_N\)-invariant, \(i\)-invariant subset \(L\subseteq\partial^2(F_N)\). The elements of a lamination \(X\) are called leaves and it is called arational if no leaf is carried by a proper free factor of \(F_N\).

Associated to a \(T\in\partial cv_N\) is a lamination \(L(T)\), which is constructed as follows. Let \(L_\varepsilon(T)=\overline{\{(g^{-\infty},g^\infty\mid\ell_T(g)<\varepsilon\}}\), and define \(L(T)=\bigcap_{\varepsilon>0}L_\varepsilon(T)\). Here, for a \(1\neq g\in F_N\), \(g^{-\infty}\) and \(g^\infty\) denote the attracting and repelling fixed points of \(g\) in \(\partial F_N\) and \(\ell_T\) is a length function on \(T\).

A tree \(T\in\partial CV_N\) is called arational if the lamination \(L(T)\) is arational. Let \(\mathcal{AT}\subseteq\partial CV_N\) denote the set of arational trees, equiped with the subspace topology. Define a relation \(\sim\) on \(\mathcal{AT}\) by \(S\sim T\) if and only if \(L(S)=L(T)\), and give \(\mathcal{AT}/\sim\) the quotient topology.

Now we can state the main theorem. Theorem. The space \(\partial\mathcal F\) is homeomorphic to \(\mathcal{AT}/\sim\).

The arguments of the authors use the geometry of Outer space and folding paths as developed by M. Bestvina and M. Feighn [op. cit.] and the structure theory of trees in \(\partial CV_N\) developed by Coulbois, Hilion, Lustig, and Reynolds in recent series of papers.

As the authors point out the theorem above is a very strong analogy of E. Klarreich’s description, [in The boundary at infinity of the complex of curves and the relative Teichmüller space, preprint https://pressfolios-production.s3.amazonaws.com/uploads/story/story-pdf/145710/1457101434403642.pdf], of the boundary \(\partial\mathcal C(S)\) of the complex of curves \(\mathcal C(S)\) associated to a nonexceptional surface \(S\).

Reviewer: Dimitrios Varsos (Athína)

##### MSC:

20F65 | Geometric group theory |

20E05 | Free nonabelian groups |

20E08 | Groups acting on trees |

20F67 | Hyperbolic groups and nonpositively curved groups |

37A25 | Ergodicity, mixing, rates of mixing |

37B10 | Symbolic dynamics |

57M07 | Topological methods in group theory |