# Trees are nilrigid

## TL;DR

This paper proves that on regular trees, any cellular automaton that eventually stabilizes all configurations must do so in finite time, establishing a rigidity property called nilrigidity for the automorphism group actions.

## Contribution

It introduces and proves the nilrigidity of automorphism group actions on regular trees, extending the understanding of cellular automata behavior on non-Euclidean structures.

## Key findings

- Asymptotically nilpotent CA are nilpotent on regular trees.
- The full automorphism group action on the tree is nilrigid.
- Open question about existence of simply transitive nilrigid automorphism actions.

## Abstract

We study cellular automata on the unoriented $k$-regular tree $T_k$, i.e. continuous maps acting on colorings $T_k$ which commute with all automorphisms of the tree. We prove that every CA that is asymptotically nilpotent, meaning every configuration converges to the same constant configuration, is nilpotent, meaning each configuration is mapped to that configuration after finite time. We call group actions nilrigid when their cellular automata have this property, following Salo and T\"orm\"a. In this terminology, the full action of the automorphism group of the $k$-regular tree is nilrigid. We do not know whether there is a nilrigid automorphism group action on $T_k$ that is simply transitive on vertices.