Groups admitting Wirtinger presentations and Gromov hyperbolic groups

TL;DR

This paper investigates Gromov hyperbolic groups with twisted Wirtinger presentations, showing that their second rational homology vanishes, and for torsion-free groups, their second integral homology also vanishes, revealing new homological properties.

Contribution

It establishes a link between twisted Wirtinger presentations and the vanishing of second homology in Gromov hyperbolic groups, extending understanding of their algebraic topology.

Findings

Second rational homology vanishes for such groups

Second integral homology vanishes if the group is torsion-free

Provides new insights into the structure of Gromov hyperbolic groups with specific presentations

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.