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 $K$ 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 $K'$. This result follows from our proof of a conjecture of Friedl and L\"uck. Specifically, we show that any 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. Under these conditions, we also establish new upper bounds on the number of faces of the Thurston norm ball of the target manifold. To resolve Friedl and L\"uck's conjecture, we prove that locally indicable $3$-manifold groups are Lewin groups, thereby confirming a 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.