Anosov flows on -manifolds: the surgeries and the foliations

Abstract

Every Anosov flow on a 3-manifold is associated to a bifoliated plane (a plane endowed with two transverse foliations $F^s$ and $F^u$ ) which reflects the normal structure of the flow endowed with the center-stable and center-unstable foliations. A flow is $\mathbb{R}$ -covered if $F^s$ (or equivalently $F^u$ ) is trivial. On the other hand, from any Anosov flow one can build infinitely many others by Dehn–Goodman–Fried surgeries. This paper investigates how these surgeries modify the bifoliated plane. We first observe that surgeries along orbits corresponding to disjoint simple closed geodesics do not affect the bifoliated plane of the geodesic flow of a hyperbolic surface (Theorem 1). Analogously, for any non- $\mathbb{R}$ -covered Anosov flow, surgeries along pivot periodic orbits do not affect the branching structure of its bifoliated plane (Theorem 2). Next, we consider the set $\mathcal{S}urg(A)$ of Anosov flows obtained by Dehn–Goodman–Fried surgeries from the suspension flow $X_A$ of any hyperbolic matrix $A \in SL(2,\mathbb{Z})$ . Fenley proved that performing only positive (or negative) surgeries on $X_A$ leads to $\mathbb{R}$ -covered Anosov flows. We study here Anosov flows obtained by a combination of positive and negative surgeries on $X_A$ . Among other results, we build non- $\mathbb{R}$ -covered Anosov flows on hyperbolic manifolds. Furthermore, we show that given any flow $X\in \mathcal{S}urg(A)$ there exists $\epsilon>0$ such that every flow obtained from $X$ by a non-trivial surgery along any $\epsilon$ -dense periodic orbit $\gamma$ is $\mathbb{R}$ -covered (Theorem 4). Analogously, for any flow $X \in \mathcal{S}urg(A)$ there exist periodic orbits $\gamma_+,\gamma_-$ such that every flow obtained from $X$ by surgeries with distinct signs on $\gamma_+$ and $\gamma_-$ is non- $\mathbb{R}$ -covered (Theorem 5).

Constructing Birkhoff sections for pseudo-Anosov flows with controlled complexity

“…In particular, since plenty of non-R-covered Anosov flows are known to exist (see, for example [BI23]), this shows that Theorem 1.2 is sharp. This also gives an exact classification of Anosov flows for which Theorem 1.2 is sharp.…”

Constructing Birkhoff sections for pseudo-Anosov flows with controlled complexity

“…It is known that there exists non-R-covered Anosov flows, see for example [BI20]. Hence for those flows, there cannot be a Birkhoff section with only 1 boundary component.…”

Construction of Anosov flows in dimension 3 by gluing blocks

“…S. Fenley showed in [17] that any Anosov flow obtained by Dehn-Goodman-Fried surgeries of coherent orientations on a suspension or a geodesic flow is skewed R-covered. C. Bonatti and I. Iakovoglou showed in [13] that if X is an Anosov field obtained by Dehn-Goodman-Fried surgeries from a suspension then any surgery on an ϵ-dense periodic orbit of X yields a skewed R-covered Anosov flow.…”