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 Z, we show that a very weak gluing property suffices for nil-rigidity. For Z2-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.