Zippers

Calegari, Danny; Loukidou, Ino

doi:10.2140/gt.2026.30.1931

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ISSN (electronic): 1364-0380

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Zippers

Danny Calegari and Ino Loukidou

Geometry & Topology 30 (2026) 1931–1962

DOI: 10.2140/gt.2026.30.1931

Abstract

If $M$ is a hyperbolic 3-manifold fibering over the circle, the fundamental group of $M$ acts faithfully by homeomorphisms on a circle (the circle at infinity of the universal cover of the fiber), preserving a pair of invariant (stable and unstable) laminations. Many different kinds of dynamical structures (e.g., taut foliations, quasigeodesic or pseudo-Anosov flows) are known to give rise to universal circles — a circle with a faithful $π_{1} (M)$ -action preserving a pair of invariant laminations — and these universal circles play a key role in relating the dynamical structure to the geometry of $M$ . In this paper we introduce the idea of zippers, which give a new and direct way to construct universal circles, streamlining the known constructions in many cases, and giving a host of new constructions in others. In particular, zippers (and their associated universal circles) may be constructed directly from uniform quasimorphisms or from uniform actions.

Keywords

hyperbolic $3$-manifold, uniform quasimorphism, zipper, laminations, universal circle, left orders

Mathematical Subject Classification

Primary: 37C85, 57K32, 57R30

References

Publication

Received: 23 November 2024

Revised: 3 February 2026

Accepted: 4 March 2026

Published: 18 July 2026

Proposed: Ian Agol

Seconded: Leonid Polterovich, David Gabai

Authors

Danny Calegari
Department of Mathematics University of Chicago Chicago, IL United States

Ino Loukidou
Department of Mathematics University of Chicago Chicago, IL United States

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Volume 30, issue 5 (2026)