Group actions on treelike compact spaces

TL;DR

This paper demonstrates that group actions on various treelike compact spaces are generally simple and tame, using classical and new methods, with implications for group actions on dendrites and related structures.

Contribution

It shows that all actions of topological groups on treelike compact spaces like dendrons are null or tame, extending classical results and introducing new methods for these analyses.

Findings

Group actions on regular continua are null and tame.

Actions on local dendrons are null.

Actions on dendrons are Rosenthal representable and tame.

Abstract

We show that group actions on many treelike compact spaces are not too complicated dynamically. We first observe that an old argument of Seidler implies that every action of a topological group $G$ on a regular continuum is null and therefore also tame. As every local dendron is regular, one concludes that every action of $G$ on a local dendron is null. We then use a more direct method to show that every continuous group action of $G$ on a dendron is Rosenthal representable, hence also tame. Similar results are obtained for median pretrees. As a related result we show that Helly's selection principle can be extended to bounded monotone sequences defined on median pretrees (e.g., dendrons or linearly ordered sets). Finally, we point out some applications of these results to continuous group actions on dendrites.