Domination Invariance and Bounding Geometries in O-minimal Expansions of Ordered Abelian Groups

We study global definable types satisfying joint elementary end-extensionconditions on positive definable cones in o-minimal expansions of divisible orderedabelian groups. We establish pushforward heredity under partial definable maps andisolate a field obstruction in the real-closed-field setting. In the non-field case, weshow that pure-long domination types introduce no new convex-bounded elementsover a base model, while every definable type carrying the short domination directionfails the joint end condition on every positive-dimensional cone. Under a convex-boundedness hypothesis on the additional bounding relations, these two mechanismsyield an exact classification: the proper global definable joint end-types are preciselythe types in the pure-long domination class. Consequently, within this geometricregime, the joint end-type property is invariant under domination equivalence. Wealso derive a 1-type factorization criterion for joint end-types. The final sectionidentifies the remaining natural-closure and cofinal-curve questions leading to furtherwork.