Commutators cannot be proper powers in metric small-cancellation torsion-free groups

TL;DR

This paper proves that in certain torsion-free groups with small cancellation properties, nontrivial commutators cannot be expressed as proper powers, highlighting a structural restriction in these groups.

Contribution

It establishes a new restriction on the algebraic structure of torsion-free small-cancellation groups by ruling out proper powers among nontrivial commutators.

Findings

Nontrivial commutators are not proper powers in these groups.

The result applies to groups satisfying C'( extstyle{rac{1}{6}}) small cancellation condition.

Provides insight into the algebraic constraints of small-cancellation torsion-free groups.

Abstract

A nontrivial commutator cannot be a proper power in a torsion-free group satisfying C'(\lambda) small cancellation condition with sufficiently small \lambda.