New Anosov flows via bicontact structures

TL;DR

This paper introduces a novel method for constructing hyperbolic flows on 3-manifolds using bicontact structures, enabling the creation of new transitive Anosov flows on complex manifolds.

Contribution

It develops a new approach to hyperbolic plugs via bicontact structures, allowing the construction of closed manifolds with diverse Anosov flows and relating surgeries to flow equivalences.

Findings

Constructed new transitive Anosov flows on surface bundle manifolds.

Demonstrated manifolds with multiple nonequivalent Anosov flows.

Connected generalized surgeries to sequences of Goodman--Fried surgeries.

Abstract

We present a new approach to hyperbolic plugs, via a construction of bicontact plugs on 3-manifolds with boundary that are surface bundles over the circle. The boundary components are quasi transverse tori, and we prove a gluing theorem that allows us to produce closed manifolds carrying new transitive Anosov flows. We show that a toroidal manifold produced by gluing two copies of the figure eight knot complement may carry many nonequivalent Anosov flows, and likewise a manifold composed of a figure eight complement and a trefoil complement. We further show that certain generalized Handel--Thurston surgeries can be realized as sequences of Goodman--Fried surgeries and produce new examples of different surgery sequences resulting in the same Anosov flow.