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Rips induction: index of the dual lamination of an \(\mathbb R\)-tree. (English) Zbl 1336.20033
Summary: Let \(T\) be an \(\mathbb R\)-tree in the boundary of the outer space \(\mathrm{CN}_N\), with dense orbits. The \(\mathcal Q\)-index of \(T\) is defined by means of the dual lamination of \(T\). It is a generalisation of the Poincaré Lefschetz index of a foliation on a surface. We prove that the \(\mathcal Q\)-index of \(T\) is bounded above by \(2N-2\), and we study the case of equality. The main tool is to develop the Rips machine in order to deal with systems of isometries on compact \(\mathbb R\)-trees. Combining our results on the \(\mathcal Q\)-index with results on the classical geometric index of a tree, developed by D. Gaboriau and G. Levitt [Ann. Sci. Éc. Norm. Supér. (4) 28, No. 5, 549-570 (1995; Zbl 0835.20038)], we obtain a beginning classification of trees.

MSC:
20E08 Groups acting on trees
20E05 Free nonabelian groups
20F65 Geometric group theory
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References:
[1] M. Bestvina and M. Feighn, Outer limits. Unpublished, 1994.
[2] M. Bestvina and M. Feighn, Stable actions of groups on real trees. Invent. Math. 121 (1995), 287-321. · Zbl 0837.20047
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