Approximations of stable actions on $ \Bbb {R} $-trees

Guirardel, Vincent

doi:10.1007/s000140050047

Commentarii Mathematici Helvetici

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Volume 73, Issue 1, 1998, pp. 89–121

DOI: 10.1007/s000140050047

Published online: 1998-03-31

Approximations of stable actions on $ \Bbb {R} $-trees

Vincent Guirardel^[1]

(1) Université de Rennes 1, Rennes, France

This article shows how to approximate a stable action of a finitely presented group on an $ \Bbb {R} $-tree by a simplicial one while keeping control over arc stabilizers. For instance, every small action of a hyperbolic group on an $ \Bbb {R} $-tree can be approximated by a small action of the same group on a simplicial tree. The techniques we use highly rely on Rips's study of stable actions on $ \Bbb {R} $-trees and on the dynamical study of exotic components by D. Gaboriau.

Keywords: Stable action, $ \Bbb {R} $$ -trees, Rips theorem, approximation, splitting, Bass-Serre

Guirardel Vincent: Approximations of stable actions on $ \Bbb {R} $-trees. Comment. Math. Helv. 73 (1998), 89-121. doi: 10.1007/s000140050047