The talk is based on the joint work with Cornelia Drutu, Alexander Olshanskii and Denis Osin. Many questions about automorphisms and endomorphisms of groups, JSJdecompositions, the Hopf and co-Hopf properties, solutions of equations in groups, etc. lead to actions of groups on asymptotic cones of themselves or asymptotic cones of other groups. In the hyperbolic case this leads to the Rips-Sela theory. We show that it can be extended to a very large class of groups whose asymptotic cones have cut points.