Left-orderability in Dehn fillings of pseudo-Anosov mapping tori

Abstract

For pseudo-Anosov mapping tori with co-orientable invariant foliations and monodromies reversing their co-orientations, a family of taut foliations was constructed in previous work on Dehn fillings with all rational slopes outside a neighborhood of the degeneracy slope. In this paper, we prove that all such Dehn fillings have left-orderable fundamental groups. We present two approaches, both establishing left-orderability through the branching behavior of taut foliations. The first approach produces an $\mathbb{R}$-covered foliation arising from this family for each filling slope, and the second approach shows that, depending on the choice of a suitable system of arcs on $\Sigma$, the resulting foliation either has one-sided branching or is $\mathbb{R}$-covered. Consequently, the second approach associates to each Dehn filling a family of representations of its fundamental group into ${\mathcal{G}}_{\infty }$, the group of germs at infinity, whereas the first approach yields an explicit left-invariant order. As an application, combining our results with earlier work in the literature, we verify the L-space conjecture for all surgeries on the $(-2,3,2q+1)$-pretzel knot ($q⩾3$) in $S^3$. From another perspective, $\mathbb{R}$-covered foliations can be produced systematically across a large family of Dehn fillings on cusped hyperbolic manifolds, and in some cusped manifolds they cover all fillings that admit co-orientable taut foliations. This expands the class of known $\mathbb{R}$-covered foliations.

31 pages, 27 figures