The action on a tree associated to an ideal point of the character variety

Let G be the fundamental group of a compact, orientable, irreducible 3-manifold M. I will discuss the correspondence between ideal points of a curve in the character variety X(G) and actions of G on a tree (following Culler-Shalen). This is a particular case of Bass-Serre theory concerning subgroups of SL_2(K), where K is a local field. I will spend some time discussing the tree for SL_2(K) and its relation to graphs of groups, and time permitting, how this relates to the existence of incompressible surfaces in M.