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.

30 pages, 6 figures; incorporates feedback from referee

References in corpus (7)

• Holomorphic disks and genus bounds
• Laminations and groups of homeomorphisms of the circle
• Ideal boundaries of pseudo-Anosov flows and uniform convergence groups, with connections and applications to large scale geometry
• Universal circles for quasigeodesic flows
• Floer homology, group orderability, and taut foliations of hyperbolic 3-manifolds
• Quasigeodesic flows and sphere-filling curves
• Endperiodic maps, splitting sequences, and branched surfaces