On knot groups acting on trees

TL;DR

This paper characterizes 1-knot groups as generalized Baumslag--Solitar groups, showing they are precisely torus-knot groups, and extends the classification to higher-dimensional knots for n ≥ 3.

Contribution

It provides a complete characterization of 1-knot groups as GBS groups and describes all n-knot GBS groups for dimensions n ≥ 3.

Findings

1-knot groups are exactly torus-knot groups within GBS groups

All n-knot GBS groups for n ≥ 3 are classified

The paper links knot groups to actions on trees with cyclic stabilizers

Abstract

A finitely generated group $G$ acting on a tree with infinite cyclic edge and vertex stabilizers is called a generalized Baumslag--Solitar group ($GBS$ group). We prove that a 1-knot group $G$ is $GBS$ group iff $G$ is a torus-knot group and describe all n-knot GBS groups for $n\geqslant 3$.