Groups admitting Wirtinger presentations and Gromov hyperbolic groups

TL;DR

The paper identifies a homological obstruction for finitely generated groups admitting twisted Wirtinger presentations to be Gromov hyperbolic. It leverages a result of Kuzmin showing that any element of $H_2(G)$ comes from a wedge of commuting elements, and then argues that absence of a $\mathbb{Z}\times\mathbb{Z}$ subgroup forces $H_2(G;\mathbb{Q})=0$ (and $H_2(G;\mathbb{Z})=0$ when torsion-free). Since Gromov hyperbolic groups do not contain $\mathbb{Z}\times\mathbb{Z}$, they satisfy the vanishing obstruction, excluding many twisted Wirtinger groups from being hyperbolic. The results highlight second homology as a diagnostic tool distinguishing hyperbolic from certain non-hyperbolic twisted Wirtinger groups.

Abstract

Twisted Wirtinger presentations are generalizations of the classical Wirtinger presentations of knot and link groups. In this paper, we prove that if a finitely generated group admitting a twisted Wirtinger presentation is Gromov hyperbolic, then its second rational homology group vanishes. Moreover, if the group is torsion-free, then its second integral homology group also vanishes.

Theorems & Definitions (4)

• Theorem 1
• Corollary
• Theorem 2: Kuz min MR1392843
• proof : Proof of Theorem \ref{['thm:algebra']}