Let be a -tree in the boundary of the Outer Space CV, with dense orbits. The -index of is defined by means of the dual lamination of . It is a generalisation of the Poincaré Lefschetz index of a foliation on a surface. We prove that the -index of is bounded above by , 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 -trees.
Combining our results on the -index with results on the classical geometric index of a tree, developed by Gaboriau and Levitt, we obtain a beginning classification of trees.
Cite this article
Thierry Coulbois, Arnaud Hilion, Rips induction: index of the dual lamination of an -tree. Groups Geom. Dyn. 8 (2014), no. 1, pp. 97–134DOI 10.4171/GGD/218