Left orderability and taut foliations with orderable cataclysm

Abstract

Let $M$ be a connected, closed, orientable, irreducible $3$-manifold. We show that: if $M$ admits a co-orientable taut foliation $\mathcal{F}$ with orderable cataclysm, then $π_1(M)$ is left orderable. This provides an elementary proof that $π_1(M)$ is left orderable if $M$ admits an Anosov flow with a co-orientable stable foliation without using Thurston's universal circle action. Furthermore, for every closed orientable 3-manifold that admits a pseudo-Anosov flow $X$ with a co-orientable stable foliation, our result applies to infinitely many of Dehn fillings along the union of singular orbits of $X$.

Figures (7)

• Figure 1: Suppose that $L = L(\mathcal{F})$ for some taut foliation $\mathcal{F}$ and $L$ is non-Hausdorff. Picture (a) is the local model of a cataclysm $\mu$ of $L$. Fix an orientation on $L$, picture (b) describes a positive end and a negative end of $L$: the positive end can be represented by a positively oriented ray of $L$, and the negative end can be represented by a negatively oriented ray of $L$.
• Figure 2: (a) is the local model of an upward cataclysm in $L$ and (b) is the local model of a downward cataclysm in $L$, where $L$ is labeled with the positive orientation.
• Figure 3: We give an example of $\alpha(u,v)$ (where $u = t_1$ and $v = t_{16}$ in the picture). The sequence of points $\{t_i\}_{1 \leqslant i \leqslant 16}$ is shown in picture (a), where $t_9 = t_{10}$. The paths $\{\gamma_k\}_{1 \leqslant k \leqslant 8}$ and cataclysms $\{\mu_k\}_{1 \leqslant k \leqslant 7}$ are shown in picture (b), where $\gamma_5$ is a trivial path. In the sequence $\gamma_1,\mu_1,\gamma_2,\ldots,\mu_7,\gamma_8$, all $\gamma_j, \mu_k$ have the relations as given in Condition \ref{['condition 4']}.
• Figure 4: We give an example of $\alpha(x_1,x_2)$ for two distinct ends $x_1, x_2$ of $L$. In picture (a), there are two rays $r_1,r_2$ which represent $x_1, x_2$ respectively and do not intersect, and we let $u = r_1(0)$, $v = r_2(0)$. The blue broken path in picture (b) is $\alpha(u,v)$. Picture (c) describes the broken curve that goes from $x_1$ to $x_2$ along $\overline{r_1}$ (i.e. $r_1$ with inverse direction), $\alpha(u,v)$ and $r_2$, which has some self-intersections. The red broken curve in picture (d) is $\alpha(x_1,x_2)$, obtained from deleting the self-intersections of the previous broken curve.
• Figure 5: The pictures of $\beta_1$, $\beta_2$, $\beta_3$. Here, the "common intersection place" of $\beta_1$, $\beta_2$, $\beta_3$ is undetermined (it has more than one cases).

Theorems & Definitions (38)

• Conjecture 1: L-space conjecture
• Theorem 1.1
• Corollary 1.2
• Remark 1.3
• Remark 1.4
• Proposition 1.5
• Remark 1.6
• Definition 2.2
• Definition 2.3
• Definition 2.4