Abstract:

Given two length functions $\ell, m$ of minimal irreducible G actions on R-trees A, B, when is $\ell + m$ again the length function of a minimal irreducible $G$ action on an R-tree? We will show that additivity is characterized by the geometry of the Guirardel core of A and B, and also by a combinatorial compatibility condition generalizing the condition given by Behrstock, Bestvina, and Clay for F_n actions on simplicial trees. This compatibility condition allows us to characterize the PL-geometry of common deformation spaces of R-trees, such as the closure of Culler-Vogtmann Outer Space or the space of small actions of a hyperbolic group G. Time permitting application of this theory to understanding two-generator subgroups of Out(F_n) will be discussed.