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Ergodic currents dual to a real tree. (English) Zbl 1362.37085
Let \(F_{N}\) 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 \(N\).

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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