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Products of loxodromic automorphisms of pretrees. (English) Zbl 1267.20043

From the introduction: According to M. Culler and J. W. Morgan [Proc. Lond. Math. Soc., III. Ser. 55, 571-604 (1987; Zbl 0658.20021)], if \(h_1\) and \(h_2\) are hyperbolic isometries of an \(\mathbb R\)-tree, then one of the following properties holds: (a) the axis of \(h_1\) meets the axis of \(h_2\) and \(\max(|h_2h_1|,|h_2^{-1}h_1|)=|h_1|+|h_2|\); (b) the axis of \(h_1\) does not meet the axis of \(h_2\) and \(|h_2h_1|=|h_2^{-1}h_1|>|h_1|+|h_2|\) (here \(|h|\) is the hyperbolic length of \(h\)).
When one considers spaces more general than trees and actions more general than isometries (for example non-nesting homeomorphisms of an \(\mathbb R\)-tree) properties of this kind (may be in a different form) are very helpful. We prove a variant of this property in a very general situation when \(h_1\) and \(h_2\) are automorphisms of a median pretree.

MSC:

20E08 Groups acting on trees

Citations:

Zbl 0658.20021
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References:

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