Ramification Subgroups of Knot Groups and their Profinite and Cohomological Structure

TL;DR

This paper develops a ramification theory for knot group covers, analyzing the structure of ramification subgroups through finite quotients, profinite completions, and cohomology, revealing parallels with number theory.

Contribution

It introduces a formal ramification framework for knot exteriors, connecting algebraic, topological, and cohomological perspectives in a novel way.

Findings

U/M_U is the universal maximal meridionally unramified quotient.

Profinite ramification subgroup is generated by inertia.

Unramified H^1-classes vanish on inertia subgroups.

Abstract

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.