The Lipschitz metric on deformation spaces of $G$-trees

Abstract

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 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 $BS(1,n)$ with $n\geq 2$.