Confined subgroups and high transitivity

TL;DR

This paper investigates the structure of highly transitive group actions, showing that confined subgroups inherit high transitivity under certain conditions, aiding in classifying such actions especially in dynamical groups.

Contribution

It establishes that confined subgroups of highly transitive groups retain high transitivity, providing a new method to classify and rule out certain highly transitive actions.

Findings

Confined subgroups of highly transitive groups are also highly transitive after discarding finitely many points.

The result helps classify highly transitive actions of finitely generated groups.

Application to groups of dynamical origin yields the first non-trivial classification.

Abstract

An action of a group $G$ is highly transitive if $G$ acts transitively on $k$-tuples of distinct points for all $k \geq 1$. Many examples of groups with a rich geometric or dynamical action admit highly transitive actions. We prove that if a group $G$ admits a highly transitive action such that $G$ does not contain the subgroup of finitary alternating permutations, and if $H$ is a confined subgroup of $G$, then the action of $H$ remains highly transitive, possibly after discarding finitely many points. This result provides a tool to rule out the existence of highly transitive actions, and to classify highly transitive actions of a given group. We give concrete illustrations of these applications in the realm of groups of dynamical origin. In particular we obtain the first non-trivial classification of highly transitive actions of a finitely generated group.