#### Volume 15, issue 2 (2015)

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

### Sebastian Meinert

Algebraic & Geometric Topology 15 (2015) 987–1029
##### 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 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 $BS\left(1,n\right)$ with $n\ge 2$.

##### Keywords
Lipschitz metric, deformation spaces, $G$–trees, outer automorphisms, train tracks, virtually free groups, generalized Baumslag–Solitar groups
##### Mathematical Subject Classification 2010
Primary: 20F65, 20E08
Secondary: 20E36
##### Publication
Received: 9 May 2014
Revised: 20 September 2014
Accepted: 25 September 2014
Published: 22 April 2015
##### Authors
 Sebastian Meinert Freie Universität Berlin Institut für Mathematik Arnimallee 7, 14195 Berlin Germany http://meinert-online.de/personal/