Ergodic currents dual to a real tree.

*(English)*Zbl 1362.37085Let \(F_{N}\) be the free group of rank \(N\) and \(\partial F_{N}\) the Gromov boundary of \(F_{N}\). The action of \(F_{N}\) on itself by left multiplication extends continuously to \(\partial F_{N}\). Let \(\partial ^{2}F_{N}\,=\,(\partial F_{N})^{2}\smallsetminus \Delta\), where \(\Delta \) stands for the diagonal. \(\partial ^{2}F_{N}\) inherits a product topology from \(\partial F_{N}\) and the action of \(F_{N}\) on \(\partial F_{N}\) gives rise a diagonal action on \(\partial ^{2}F_{N}\). A current \(\mu \) is a \(F_{N}\)-invariant, flip-invariant, Radon measure on \(\partial ^{2}F_{N}\) (namely a Borel measure which is finite on compact sets). The space \(\mathrm{Curr}(F_{N})\) of currents of \(F_{N}\) is equipped with the weak-\(\ast \)-topology: it is a locally compact space. The space \(\mathbb{P}\mathrm{Curr}(F_{N})\) of projective classes of (non-zero) currents equipped with the quotient topology is a compact set. The group \(\mathrm{Out}(F_{N})\) acts on \(\mathrm{Curr}(F_{N})\) (see in [I. Kapovich, Contemp. Math. 394, 149–176 (2006; Zbl 1110.20034)]) and this action induces an action on \(\mathbb{P}\mathrm{Curr}(F_{N})\).

Let \(CV_{N}\) be the outer space of projective classes of free minimal actions of \(F_{N}\) by isometries on simplicial metric trees and \(cv_{N}\) be the unprojectivized outer space (see in [M. Culler and K. Vogtmann, Invent. Math. 84, 91–119 (1986; Zbl 0589.20022)]). The space \(CV_{N}\) admits a Thurston boundary, which gives rise to a compactification \(\overline{CV_{N}}\). Similarly, a compactification \(\overline{cv_{N}}\) of \(cv_{N}\) is obtained.

In [I. Kapovich and M. Lustig, Geom. Topol. 13, No. 3, 1805–1833 (2009; Zbl 1194.20046)], and studied a kind of duality bracket is defined:

\[ \langle\,\cdot ,\,\cdot\, \rangle : \overline{cv_{N}}\times \mathrm{Curr}(F_{N})\,\longrightarrow\,\mathbb{R}_{\geq 0}. \]

The number \( \langle\,T,\,\mu\, \rangle\) is the intersection number of \(T\) and \(\mu\).

A tree \(T\in \overline{cv_{N}}\) and a current \(\mu \in \mathrm{Curr}(F_{N})\) are dual when \(\langle\,T,\,\mu \, \rangle\,=\,0\). The set of projective classes of currents dual to a given tree \(T\) is convex: the extremal points of this set are ergodic currents.

In this paper, it is explained how all the currents dual to a given tree are constructed.

Theorem A. Let \(T\) be a \(\mathbb{R}\)-tree with a minimal, free action of \(F_{N}\) by isometries with dense orbits. There are at most \(3N-5\) projective classes of ergodic currents dual to \(T\).

The duality between trees and currents can also be understood by considering laminations. An \(\mathbb{R}\)-tree \(T\) with an action of \(F_{N}\) by isometries with dense orbits has a dual lamination \(L(T)\) [the first author et al., J. Lond. Math. Soc., II. Ser. 78, No. 3, 755–766 (2008; Zbl 1200.20018)]. In [I. Kapovich and M. Lustig, Geom. Funct. Anal. 19, No. 5, 1426–1467 (2010; Zbl 1242.20052)] it is proved that a current is dual to \(T\) if and only if it is carried by the dual lamination. Thus, the theorem above can be rephrased.

Theorem B. Let \(T\) be an \(\mathbb{R}\)-tree with a minimal, free action of \(F_{N}\) by isometries with dense orbits. The dual lamination \(L(T)\) carries at most \(3N-5\) projective classes of ergodic currents.

For the proof of this theorem the authors use a kind of band complex defined in [the first author et al., Math. Proc. Camb. Philos. Soc. 147, No. 2, 345–368 (2009; Zbl 1239.20030)] and the unfolding induction, which consists in either the Rips induction [the authors, Groups Geom. Dyn. 8, No. 1, 97–134 (2014; Zbl 1336.20033)] or the splitting induction [the first author et al., Groups Geom. Dyn. 9, No. 2, 567–597 (2015; Zbl 1342.20028)].

In the case where the Rips induction completely decompose the tree, the following claim is proved:

Theorem B\(_{1}\). Let \(T\) be an \(\mathbb{R}\)-tree with a minimal, free action of \(F_{N}\) by isometries with dense orbits. Assume that \(T\) is a tree of Levitt type. Then the simplex \(\mathbb{P}\mathrm{Curr}(T)\) of currents carried by the dual lamination \(L(T)\) has dimension at most \(3N-4\) (and projective dimension at most \(3N-5\)).

There are trees in the boundary of outer space where the Rips induction is useless: the trees of surface type in [the authors, Groups Geom. Dyn. 8, No. 1, 97–134 (2014; Zbl 1336.20033)]. For these trees the authors define another kind of induction, called splitting induction (see paragraph 4 in the paper) and prove Theorem B when the induction completely analyzes the lamination.

Theorem B\(_{2}\). Let \(T\) be an \(\mathbb{R}\)-tree with a minimal free action of \(F_{N}\) by isometries with dense orbits. Let \(A\) be a basis of \(F_{N}\) and let \(S_{A}\,=\,(K_{A},\,A)\) be the associated system of isometries. Let \(S_{n}\,=\,(F_{n},\,A_{n})\) be a sequence of systems of isometries obtained from \(S_{0}\,=\,S_{A}\) by unfolding induction. Assume that each nested intersection of connected components \(u_{n}\) of \(F_{n}\) is a singleton. Then the simplex of currents \(\mathbb{P}\mathrm{Curr}(T)\) carried by the dual lamination \(L(T)\) has dimension at most \(3N-4\) (and projective dimension at most \(3N-5\)).

Since there exist \(\mathbb{R}\)-trees with no induction sequence that completely analyzes the lamination these trees are decomposable in the sense of V. Guirardel [Ann. Inst. Fourier 58, No. 1, 159–211 (2008; Zbl 1187.20020)]. In this case, however, the induction procedure ends up with a subtree with an action of a subgroup of \(F_{N}\) with rank strictly smaller than \(N\). This allows to conclude the proof of Theorem A by induction on \(N\).

Let \(CV_{N}\) be the outer space of projective classes of free minimal actions of \(F_{N}\) by isometries on simplicial metric trees and \(cv_{N}\) be the unprojectivized outer space (see in [M. Culler and K. Vogtmann, Invent. Math. 84, 91–119 (1986; Zbl 0589.20022)]). The space \(CV_{N}\) admits a Thurston boundary, which gives rise to a compactification \(\overline{CV_{N}}\). Similarly, a compactification \(\overline{cv_{N}}\) of \(cv_{N}\) is obtained.

In [I. Kapovich and M. Lustig, Geom. Topol. 13, No. 3, 1805–1833 (2009; Zbl 1194.20046)], and studied a kind of duality bracket is defined:

\[ \langle\,\cdot ,\,\cdot\, \rangle : \overline{cv_{N}}\times \mathrm{Curr}(F_{N})\,\longrightarrow\,\mathbb{R}_{\geq 0}. \]

The number \( \langle\,T,\,\mu\, \rangle\) is the intersection number of \(T\) and \(\mu\).

A tree \(T\in \overline{cv_{N}}\) and a current \(\mu \in \mathrm{Curr}(F_{N})\) are dual when \(\langle\,T,\,\mu \, \rangle\,=\,0\). The set of projective classes of currents dual to a given tree \(T\) is convex: the extremal points of this set are ergodic currents.

In this paper, it is explained how all the currents dual to a given tree are constructed.

Theorem A. Let \(T\) be a \(\mathbb{R}\)-tree with a minimal, free action of \(F_{N}\) by isometries with dense orbits. There are at most \(3N-5\) projective classes of ergodic currents dual to \(T\).

The duality between trees and currents can also be understood by considering laminations. An \(\mathbb{R}\)-tree \(T\) with an action of \(F_{N}\) by isometries with dense orbits has a dual lamination \(L(T)\) [the first author et al., J. Lond. Math. Soc., II. Ser. 78, No. 3, 755–766 (2008; Zbl 1200.20018)]. In [I. Kapovich and M. Lustig, Geom. Funct. Anal. 19, No. 5, 1426–1467 (2010; Zbl 1242.20052)] it is proved that a current is dual to \(T\) if and only if it is carried by the dual lamination. Thus, the theorem above can be rephrased.

Theorem B. Let \(T\) be an \(\mathbb{R}\)-tree with a minimal, free action of \(F_{N}\) by isometries with dense orbits. The dual lamination \(L(T)\) carries at most \(3N-5\) projective classes of ergodic currents.

For the proof of this theorem the authors use a kind of band complex defined in [the first author et al., Math. Proc. Camb. Philos. Soc. 147, No. 2, 345–368 (2009; Zbl 1239.20030)] and the unfolding induction, which consists in either the Rips induction [the authors, Groups Geom. Dyn. 8, No. 1, 97–134 (2014; Zbl 1336.20033)] or the splitting induction [the first author et al., Groups Geom. Dyn. 9, No. 2, 567–597 (2015; Zbl 1342.20028)].

In the case where the Rips induction completely decompose the tree, the following claim is proved:

Theorem B\(_{1}\). Let \(T\) be an \(\mathbb{R}\)-tree with a minimal, free action of \(F_{N}\) by isometries with dense orbits. Assume that \(T\) is a tree of Levitt type. Then the simplex \(\mathbb{P}\mathrm{Curr}(T)\) of currents carried by the dual lamination \(L(T)\) has dimension at most \(3N-4\) (and projective dimension at most \(3N-5\)).

There are trees in the boundary of outer space where the Rips induction is useless: the trees of surface type in [the authors, Groups Geom. Dyn. 8, No. 1, 97–134 (2014; Zbl 1336.20033)]. For these trees the authors define another kind of induction, called splitting induction (see paragraph 4 in the paper) and prove Theorem B when the induction completely analyzes the lamination.

Theorem B\(_{2}\). Let \(T\) be an \(\mathbb{R}\)-tree with a minimal free action of \(F_{N}\) by isometries with dense orbits. Let \(A\) be a basis of \(F_{N}\) and let \(S_{A}\,=\,(K_{A},\,A)\) be the associated system of isometries. Let \(S_{n}\,=\,(F_{n},\,A_{n})\) be a sequence of systems of isometries obtained from \(S_{0}\,=\,S_{A}\) by unfolding induction. Assume that each nested intersection of connected components \(u_{n}\) of \(F_{n}\) is a singleton. Then the simplex of currents \(\mathbb{P}\mathrm{Curr}(T)\) carried by the dual lamination \(L(T)\) has dimension at most \(3N-4\) (and projective dimension at most \(3N-5\)).

Since there exist \(\mathbb{R}\)-trees with no induction sequence that completely analyzes the lamination these trees are decomposable in the sense of V. Guirardel [Ann. Inst. Fourier 58, No. 1, 159–211 (2008; Zbl 1187.20020)]. In this case, however, the induction procedure ends up with a subtree with an action of a subgroup of \(F_{N}\) with rank strictly smaller than \(N\). This allows to conclude the proof of Theorem A by induction on \(N\).

Reviewer: Dimitrios Varsos (Athína)

##### MSC:

37E25 | Dynamical systems involving maps of trees and graphs |

37B05 | Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.) |

20E08 | Groups acting on trees |

37A30 | Ergodic theorems, spectral theory, Markov operators |

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