Asymptotic shape for subadditve processes on groups of polynomial growth

This study delves into the exploration of the limiting shape theorem for subadditive processes on finitely generated groups with polynomial growth, commonly referred to as virtually nilpotent groups. Investigating the algebraic structures underlying these processes, we present a generalized form of the asymptotic shape theorem within this framework. Extending subadditive ergodic theory in this context, we consider processes which exhibit both at most and at least linear random growth. We conclude with applications and illustrative examples.