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Simon's knot genus problem and Lewin $3$-manifold groups

Pablo Sánchez-Peralta

Abstract

We provide a positive answer to an old problem of Jonathan K. Simon: if $K$ and $K'$ are two knots such that there is an epimorphism from the knot group of $K$ to the knot group of $K'$, then the genus of $K$ is greater than or equal to the genus of $K'$. We achieve this by proving a conjecture of Friedl and Lück, which states that the existence of a map between admissible $3$-manifolds that induces an epimorphism on the fundamental groups and an isomorphism on the rational homologies yields an inequality of Thurston norms. We resolve Friedl and Lück's conjecture by showing that locally indicable $3$-manifold groups are Lewin groups, which confirms another conjecture of Jaikin-Zapirain within the class of $3$-manifold groups. As a further consequence of our methods, we show that the crossed product of a division ring and a torsion-free $3$-manifold group that is virtually free-by-cyclic is a pseudo-Sylvester domain.

Simon's knot genus problem and Lewin $3$-manifold groups

Abstract

We provide a positive answer to an old problem of Jonathan K. Simon: if and are two knots such that there is an epimorphism from the knot group of to the knot group of , then the genus of is greater than or equal to the genus of . We achieve this by proving a conjecture of Friedl and Lück, which states that the existence of a map between admissible -manifolds that induces an epimorphism on the fundamental groups and an isomorphism on the rational homologies yields an inequality of Thurston norms. We resolve Friedl and Lück's conjecture by showing that locally indicable -manifold groups are Lewin groups, which confirms another conjecture of Jaikin-Zapirain within the class of -manifold groups. As a further consequence of our methods, we show that the crossed product of a division ring and a torsion-free -manifold group that is virtually free-by-cyclic is a pseudo-Sylvester domain.

Paper Structure

This paper contains 10 sections, 16 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Lewin groups
  3. Preliminaries
  4. Division ring embeddings
  5. L²-Betti numbers
  6. Pseudo-Sylvester domains
  7. Locally indicable 3-manifold groups are Lewin
  8. Torsion-free virtually free-by-cyclic 3-manifold groups
  9. Locally indicable 3-manifold groups
  10. Inequality of the Thurston norm

Key Result

Theorem 1.2

Let $f \colon M \to N$ be a map between admissible $3$-manifolds which induces an epimorphism on the fundamental groups and an isomorphism $f_{*} \colon H_n(M; \mathbb{Q}) \to H_n(N; \mathbb{Q})$ for every non-negative integer $n$. Then for every $\phi \in H^1(N; \mathbb{R})$ it holds that where $f^{*} \phi$ is the pullback of $\phi$ with $f$.

Theorems & Definitions (32)

  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Conjecture 2.2: The Strong Atiyah Conjecture over $k \subseteq \mathbb{C}$
  • Theorem 2.3: cohn06FIR
  • Theorem 3.1: HennekeLopez_pseudoSylvester
  • Proposition 3.2
  • proof
  • ...and 22 more