On quasi-2-transitive actions of branch groups

TL;DR

This paper proves that branch groups, a class of automorphism groups of rooted trees, cannot have quasi-2-transitive actions on infinite sets, revealing limitations in their symmetry properties.

Contribution

It establishes a new restriction on the actions of branch groups, specifically ruling out quasi-2-transitive actions on infinite sets.

Findings

Branch groups cannot act quasi-2-transitively on infinite sets.

The result constrains the symmetry actions possible for branch groups.

Provides insight into the structure and limitations of automorphism groups of rooted trees.

Abstract

An action of a group $G$ on a set $X$ is said to be quasi-n-transitive if the diagonal action of $G$ on $X^n$ has only finitely many orbits. We show that branch groups, a special class of groups of automorphisms of rooted trees, cannot act quasi-2-transitively on infinite sets.