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\(\mathbb R\)-trees, dual laminations and compact systems of partial isometries. (English) Zbl 1239.20030
From the introduction: Let \(F_N\) be a free group of finite rank \(N\geq 2\), and let \(T\) be an \(\mathbb R\)-tree with a very small, minimal action of \(F_N\) with dense orbits. For any basis \(\mathcal A\) of \(F_N\) there exists a ‘heart’ \(K_{\mathcal A}\subset\overline T\) (= the metric completion of \(T\)) which is a compact subtree that has the property that the dynamical system of partial isometries \(a_i\colon K_{\mathcal A}\cap a_iK_{\mathcal A}\to a^{-1}_iK_{\mathcal A}\cap K_{\mathcal A}\), for each \(a_i\in\mathcal A\), defines a tree \(T_{(K_{\mathcal A},\mathcal A)}\) which contains an isometric copy of \(T\) as minimal subtree.
Theorem 1.1. Let \(T\) be an \(\mathbb R\)-tree provided with a very small, minimal, isometric action of the free group \(F_N\) with dense orbits. Let \(\mathcal A\) be a basis of \(F_N\). Then there exists a unique compact subtree \(K_{\mathcal A}\subset\overline T\) (called the ‘heart’ of \(T\) w.r.t. \(\mathcal A\)), such that for any compact subtree \(K\) of \(\overline T\) one has: \(T=T^{\min}_{\mathcal K}\Leftrightarrow K_{\mathcal A}\subseteq K\).
This is a slightly simplified version of Theorem 5.4 proved in this paper. The main tool for this proof (and indeed for the definition of the heart \(K_{\mathcal A}\)) is the dual lamination \(L(T)\). We define in this article (see Section 3) a second admissible lamination \(L_{\text{adm}}(\mathcal K)\) associated to the system of partial isometries \(\mathcal K=(K,\mathcal A)\). One key ingredient in the equivalence of Theorem 1.1 is to prove that the two statements given there are equivalent to the equation \(L(T)=L_{\text{adm}}(\mathcal K)\). The other key ingredient, developed in Section 4, is a new understanding of the crucial map \(\mathcal Q\colon\partial F_N\to\overline T\cup\partial T\) of G. Levitt and M. Lustig [J. Inst. Math. Jussieu 2, No. 1, 59-72 (2003; Zbl 1034.20038)], based on the dynamical system \(\mathcal K=(K,\mathcal A)\). The proof of Theorem 5.4 uses the full strength of the duality between trees and laminations, and in particular a transition between the two, given by the main result of our earlier paper [Ill. J. Math. 51, No. 3, 897-911 (2007; Zbl 1197.20020)].

MSC:
20E08 Groups acting on trees
20E05 Free nonabelian groups
20F65 Geometric group theory
37B10 Symbolic dynamics
57M07 Topological methods in group theory
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