2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/107529Tree-graded spaces are generalizations of R-trees. They appear as asymptotic cones of groups (when the cones have cut points). Since many questions about endomorphisms and automorphisms of groups, solving equations over groups, studying embeddings of a group into another group, etc. lead to actions of groups on the asymptotic cones, it is natural to consider actions of groups on tree-graded spaces. We develop a theory of such actions which generalizes the well known theory of groups acting on R-trees. As applications of our theory, we describe, in particular, relatively hyperbolic groups with infinite groups of outer automorphisms, and co-Hopfian relatively hyperbolic groups.01/30/06 The paper is submitted. 53 pages; Implemented several suggestions from the readers, added some resultsGroup TheoryGeometric Topology05C25, 20A15, 20E08, 20DF05, 20F15, 20F28, 20F32, 20F38, 20F65, 20F67, 20F69text