an:06305384
Zbl 1336.20033
Coulbois, Thierry; Hilion, Arnaud
Rips induction: index of the dual lamination of an \(\mathbb R\)-tree
EN
Groups Geom. Dyn. 8, No. 1, 97-134 (2014).
00333455
2014
j
20E08 20E05 20F65
\(\mathbb R\)-trees; outer space; Rips induction; dual lamination
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 Poincar?? 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 \textit{D. Gaboriau} and \textit{G. Levitt} [Ann. Sci. ??c. Norm. Sup??r. (4) 28, No. 5, 549-570 (1995; Zbl 0835.20038)], we obtain a beginning classification of trees.
Zbl 0835.20038