# zbMATH — the first resource for mathematics

$$\mathbb R$$-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
Full Text:
##### References:
  DOI: 10.1007/s002220050172 · Zbl 0887.20017  DOI: 10.1090/S0894-0347-96-00205-6 · Zbl 0877.57002  DOI: 10.1016/0040-9383(91)90002-L · Zbl 0726.57001  DOI: 10.1353/ajm.1997.0003 · Zbl 0878.20019  DOI: 10.1017/S1474748003000033 · Zbl 1034.20038  DOI: 10.1112/jlms/jdn053 · Zbl 1198.20023  DOI: 10.1007/BF02773004 · Zbl 0824.57001  DOI: 10.1112/jlms/jdn052 · Zbl 1197.20019  Gaboriau, Ann. Sci. École Norm. Sup. 28 pp 549– (1995)  Coulbois, Illinois J. Math. 51 pp 897– (2007)  Gaboriau, Ann. Inst. Fourier (Grenoble) 47 pp 101– (1997) · Zbl 0861.20030  DOI: 10.1112/plms/s3-55.3.571 · Zbl 0658.20021  DOI: 10.1016/0040-9383(94)00038-M · Zbl 0844.20018  Bowditch, Mem. Amer. Math. Soc. 662 (1999)  DOI: 10.1017/S0143385703000488 · Zbl 1076.37022  DOI: 10.2307/121043 · Zbl 0984.20025  DOI: 10.1007/BF01884300 · Zbl 0837.20047  DOI: 10.1007/BF01394344 · Zbl 0673.57034
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.