Eclipses on Zippers

TL;DR

This paper investigates the action of hyperbolic 3-manifold groups on zippers, revealing a fixed point dichotomy that advances understanding of their boundary actions and answers prior questions.

Contribution

It establishes a fixed point dichotomy for group actions on zippers, providing new insights into the boundary dynamics of hyperbolic 3-manifold groups.

Findings

Every nontrivial element fixes a unique point in each tree or acts freely on both.

There exists an element with exactly one fixed point in each tree.

The fixed point dichotomy answers a question posed by Calegari and Loukidou.

Abstract

Calegari and Loukidou introduced zippers, consisting of a disjoint pair of invariant real trees in the boundary of a closed hyperbolic 3-manifold group $\pi_1(M)$, which ensure the existence of a universal circle. We study the action of $\pi_1(M)$ on a minimal zipper and prove a fixed point dichotomy: every nontrivial element either fixes a unique point in each tree or acts freely on both. This answers a question of Calegari and Loukidou. As a consequence, there exists an element with exactly one fixed point in each tree.