Zippers

TL;DR

This paper introduces 'zippers', a novel method for constructing universal circles in hyperbolic 3-manifolds, simplifying existing processes and enabling new constructions from quasimorphisms and left orders.

Contribution

The paper presents zippers as a new technique to construct universal circles, streamlining previous methods and broadening the scope of constructions from various dynamical and algebraic structures.

Findings

Zippers provide a direct construction of universal circles.

Zippers simplify existing constructions in hyperbolic 3-manifolds.

Universal circles can be built from quasimorphisms and left orders using zippers.

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 $\pi_1(M)$ 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 left orders.