Left-orderability in Dehn fillings of pseudo-Anosov mapping tori
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918429997400064 |
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| author | Zhao, Bojun |
| author_facet | Zhao, Bojun |
| contents | 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 based on an analysis of the branching behavior from such 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 $Σ$, one obtains a foliation that 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 $(-2,3,2k+1)$-pretzel knot ($k \geqslant 3$) in $S^3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_04629 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Left-orderability in Dehn fillings of pseudo-Anosov mapping tori Zhao, Bojun Geometric Topology 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 based on an analysis of the branching behavior from such 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 $Σ$, one obtains a foliation that 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 $(-2,3,2k+1)$-pretzel knot ($k \geqslant 3$) in $S^3$. |
| title | Left-orderability in Dehn fillings of pseudo-Anosov mapping tori |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2604.04629 |