Topology and Geometry of Deformation Spaces of G-trees

Abstract

For a finitely generated group G, we study deformation spaces of metric G-trees, which are analogues of the Teichmuller spaces of surfaces for group actions on trees. Deformation spaces of metric G-trees generalize Culler-Vogtmann’s Outer space, the deformation space of free actions on trees, which has proven to be immensely useful in the study of Out(Fn), the outer automorphism group of the free group of rank n ≥ 2. Let D be a deformation space of metric G-trees. The group of positive real numbers R>0 acts on D by scaling the metrics on the trees and we define the projectivized deformation space as the quotient PD = D/R>0. The outer auto- morphism group Out(G) contains a certain subgroup OutD(G) that acts on D and PD by precomposing the G-actions on the trees. In Chapter 1, we present a complete argument that under certain assumptions the projectivized deformation space PD is a model for the classifying space of OutD(G) for a family of subgroups. In Chapter 2, we introduce an asymmetric pseudometric on PD that generalizes the asymmetric Lipschitz metric on Outer space and is an analogue of the Thurston metric on Teichmuller space. Making use of the Lipschitz metric on PD, 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 ≥ 2. In Chapter 3, we define the higher holomorphs Aut(G, k), k ∈ N, which are “higher-pointed” variants of the automorphism group Aut(G). Following the construction of the spine of Outer space, we construct a family of simplicial complexes S(PD, k), k ∈ N on which certain subgroups AutD(G, k) ≤ Aut(G, k) act and we show that these complexes are always contractible.