New Anosov flows via bicontact structures

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.

Figures (11)

• Figure 1: A Bicontact structure and the invariant foliations along a flowline of an Anosov flow.
• Figure 2: Strongly adapted bicontact structure. The blue vertical arrows represent the Reeb $R_-$ vector field of the negative contact form $\alpha_-$. The supported vector field is represented by the black arrows.
• Figure 3: The action of the bicontact surgery on a neighborhood of the surgery annulus. In blue is the plane field corresponding to $\alpha_-$ and in red the plane field of $\xi_+$.
• Figure 4: Isotopy of $\xi_+$ along the flow-lines of $R_-$. The black lines on the left are level curves of $\psi(s,v,w)$.
• Figure 5: Gluing of two copies of a bicontact plug with quasi-transverse periodic boundary $B$. The gluing map $F:B_1\rightarrow B_2$ identifies points with vertical coordinate $w$ with points $w+\frac{\pi}{n}$. Here red curves are glued to red curves and blue curves to blue curves.

Theorems & Definitions (55)

• Theorem 1.1: Gluing theorem
• Theorem 1.2: Generating bundle plugs
• Proposition 1.3: Punctured torus bundles
• Corollary 1.4
• Theorem 2.1: Hozoori Hoz5
• Remark 2.2
• Definition 3.1
• Definition 3.2
• Remark 3.3
• Definition 3.4