In this talk I will describe various aspects of our work on the real spectrum compactification of the
G-character variety X(F,G) of a finitely generated group F. This compactification is mindful of the algebraic topology of X(F,G) and contains a lot of information: for instance, points in its boundary lead canonically to F-actions on lambda buildings; these objects generalize lambda trees and affine Bruhat-Tits buildings.
When F is a compact surface group and G admits a theta positive structure, X(F,G) has connected components consisting entirely of faithful representations with discrete image, these components are the higher Teichmueller spaces. For such a component T, the mapping class group Out(F) acts with virtually abelian stabilizers on the real spectrum compactification Rspec(T) of T. This is based on a relation between Rspec(T) and the compactification of T by geodesic currents. I will explain how this relation leads to a new proof of a recent result of Charlie Reid establishing restrictions on boundary currents when G=PSL(n,R) and T is the Hitchin component. Joint work with A.Iozzi, A.Parreau, B.Pozzetti
