Torsion-free Word Hyperbolic Groups are Noncommutatively Slender

TL;DR

This paper proves that torsion-free word hyperbolic groups are noncommutatively slender, extending the class of groups known to have this property, and shows that random finitely presented groups in Gromov's sense are also noncommutatively slender.

Contribution

It establishes that torsion-free word hyperbolic groups are noncommutatively slender and that random finitely presented groups in Gromov's model share this property.

Findings

Torsion-free word hyperbolic groups are noncommutatively slender.

Random finitely presented groups in Gromov's sense are noncommutatively slender.

Abstract

In this note we prove the claim given in the title. A group G is noncommutatively slender if each map from the fundamental group of the Hawaiian Earring to G factors through projection to a canonical free subgroup. Graham Higman, in his seminal 1952 paper, proved that free groups are noncommutatively slender. Such groups were first defined by K. Eda. Eda has asked which finitely presented groups are noncommutatively slender. This result demonstrates that random finitely presented groups in the few- relator sense of Gromov are noncommutatively slender.