On the dynamics of extensions of free-abelian times free groups endomorphisms to the completion

Abstract We obtain conditions of uniform continuity for endomorphisms free-abelian times free groups the product metric defined by taking prefix in each component and establish an equivalence between this preservation a coarse-median, concept recently introduced Fioravanti. Considering extension endomorphism to completion, we count number orbits action subgroup fixed points (respectively periodic) on set infinite points. Finally, study dynamics points: automorphisms some endomorphisms, precise way, fitting classification given Delgado Ventura, prove that every point is either periodic or wandering, which implies asymptotically periodic.