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We introduce the notion of asymptotic Grothendieck groups for abelian andtriangulated categories that are both AB4 and AB4*. We study when theasymptotic Grothendieck group of the heart of a triangulated category with at-structure is isomorphic to the asymptotic Grothendieck group of thetriangulated category itself. We also explain a connexion with the notion oftopological Grothendieck group from Achar-Stroppel. Finally, we compute theasymptotic Grothendieck group of some positive (dg-)algebras having a conebounded, locally finite dimension.

Asymptotic Grothendieck groups and c.b.l.f. positive dg-algebras