  We consider expansive group actions on a compact metric space containing a
special fixed point denoted by $0$, and endomorphisms of such systems whose
forward trajectories are attracted toward $0$. Such endomorphisms are called
asymptotically nilpotent, and we study the conditions in which they are
nilpotent, that is, map the entire space to $0$ in a finite number of
iterations. We show that for a large class of discrete groups, this property of
nil-rigidity holds for all expansive actions that satisfy a natural
specification-like property and have dense homoclinic points. Our main result
in particular shows that the class includes all residually finite solvable
groups and all groups of polynomial growth. For expansive actions of the group
$\mathbb{Z}$, we show that a very weak gluing property suffices for
nil-rigidity. For $\mathbb{Z}^2$-subshifts of finite type, we show that the
block-gluing property suffices. The study of nil-rigidity is motivated by two
aspects of the theory of cellular automata and symbolic dynamics: It can be
seen as a finiteness property for groups, which is representative of the theory
of cellular automata on groups. Nilpotency also plays a prominent role in the
theory of cellular automata as dynamical systems. As a technical tool of
possible independent interest, the proof involves the construction of tiered
dynamical systems where several groups act on nested subsets of the original
space.
