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The Lipschitz metric on deformation spaces of \(G\)-trees. (English) Zbl 1364.20029

Summary: For a finitely generated group \(G\), we introduce an asymmetric pseudometric on projectivized deformation spaces of \(G\)-trees, using stretching factors of \(G\)-equivariant Lipschitz maps, that generalizes the Lipschitz metric on outer space and is an analogue of the Thurston metric on Teichmüller space. We show, that in the case of irreducible \(G\)-trees distances are always realized by minimal stretch maps, can be computed in terms of hyperbolic translation lengths and geodesics exist. We then study displacement functions on projectivized deformation spaces of \(G\)-trees and classify automorphisms of \(G\). As an application, we prove the existence of train track representatives for irreducible automorphisms of virtually free groups and nonelementary generalized Baumslag-Solitar groups that contain no solvable Baumslag-Solitar group \(\operatorname{BS}(1,n)\) with \(n\geq 2\).

MSC:

20F65 Geometric group theory
20E08 Groups acting on trees
20E36 Automorphisms of infinite groups
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