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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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##### References:
 [1] DOI: 10.1007/s002220050172 · Zbl 0887.20017 [2] DOI: 10.1090/S0894-0347-96-00205-6 · Zbl 0877.57002 [3] DOI: 10.1016/0040-9383(91)90002-L · Zbl 0726.57001 [4] DOI: 10.1353/ajm.1997.0003 · Zbl 0878.20019 [5] DOI: 10.1017/S1474748003000033 · Zbl 1034.20038 [6] DOI: 10.1112/jlms/jdn053 · Zbl 1198.20023 [7] DOI: 10.1007/BF02773004 · Zbl 0824.57001 [8] DOI: 10.1112/jlms/jdn052 · Zbl 1197.20019 [9] Gaboriau, Ann. Sci. École Norm. Sup. 28 pp 549– (1995) [10] Coulbois, Illinois J. Math. 51 pp 897– (2007) [11] Gaboriau, Ann. Inst. Fourier (Grenoble) 47 pp 101– (1997) · Zbl 0861.20030 [12] DOI: 10.1112/plms/s3-55.3.571 · Zbl 0658.20021 [13] DOI: 10.1016/0040-9383(94)00038-M · Zbl 0844.20018 [14] Bowditch, Mem. Amer. Math. Soc. 662 (1999) [15] DOI: 10.1017/S0143385703000488 · Zbl 1076.37022 [16] DOI: 10.2307/121043 · Zbl 0984.20025 [17] DOI: 10.1007/BF01884300 · Zbl 0837.20047 [18] DOI: 10.1007/BF01394344 · Zbl 0673.57034
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