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The boundary of the complex of free factors. (English) Zbl 1337.20040
Let \(F_N\) 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 \(S\).

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
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