Ramification Subgroups of Knot Groups and their Profinite and Cohomological Structure
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| Format: | Preprint |
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2026
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| author | Palaisti, Marina Pasini, Federico W. |
| author_facet | Palaisti, Marina Pasini, Federico W. |
| contents | We formalize a ramification theory for finite covers of knot exteriors. Given a knot group $G_K$ and a finite-index subgroup $U\le G_K$, we define meridional inertia subgroups $U\cap g\langle m\rangle g^{-1}$ and the global ramification subgroup $M_U\triangleleft U$ as their normal closure. We then analyze $M_U$ from three complementary viewpoints:
(1) finite quotients, where $U/M_U$ is shown to be the universal ``maximal meridionally unramified'' quotient of $U$;
(2) profinite completions, where we identify the closed ramification subgroup $\widehat M_{\widehat U}$ as the closed normal subgroup generated by closed inertia and prove that meridian-preserving isomorphisms of profinite completions preserve inertia and ramification;
(3) cohomology, where ``unramified'' $H^1$-classes (discrete and profinite) are characterized as those vanishing on all inertia subgroups, in direct analogy with number-theoretic inertia conditions in Galois cohomology. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_20365 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Ramification Subgroups of Knot Groups and their Profinite and Cohomological Structure Palaisti, Marina Pasini, Federico W. Geometric Topology Algebraic Topology Group Theory Primary 57K10, Secondary 20E18, 20J06, 11R37 We formalize a ramification theory for finite covers of knot exteriors. Given a knot group $G_K$ and a finite-index subgroup $U\le G_K$, we define meridional inertia subgroups $U\cap g\langle m\rangle g^{-1}$ and the global ramification subgroup $M_U\triangleleft U$ as their normal closure. We then analyze $M_U$ from three complementary viewpoints: (1) finite quotients, where $U/M_U$ is shown to be the universal ``maximal meridionally unramified'' quotient of $U$; (2) profinite completions, where we identify the closed ramification subgroup $\widehat M_{\widehat U}$ as the closed normal subgroup generated by closed inertia and prove that meridian-preserving isomorphisms of profinite completions preserve inertia and ramification; (3) cohomology, where ``unramified'' $H^1$-classes (discrete and profinite) are characterized as those vanishing on all inertia subgroups, in direct analogy with number-theoretic inertia conditions in Galois cohomology. |
| title | Ramification Subgroups of Knot Groups and their Profinite and Cohomological Structure |
| topic | Geometric Topology Algebraic Topology Group Theory Primary 57K10, Secondary 20E18, 20J06, 11R37 |
| url | https://arxiv.org/abs/2605.20365 |