The pure cactus group is residually nilpotent

TL;DR

This paper proves that the pure cactus group is residually nilpotent and identifies a surjective homomorphism with a residually torsion-free nilpotent kernel, advancing understanding of its algebraic structure.

Contribution

It establishes the residual nilpotency of the pure cactus group and constructs a specific surjective homomorphism with a residually torsion-free nilpotent kernel.

Findings

Pure cactus group is residually nilpotent.

Constructed a surjective homomorphism with a residually torsion-free nilpotent kernel.

Provides algebraic insights into the structure of the pure cactus group.

Abstract

We show that the pure cactus group $\Gamma_{n+1}$ is residually nilpotent and exhibit a surjective homomorphism $\Gamma_{n+1} \to (\mathbb{Z}/2\mathbb{Z})^{2^n-n(n+1)/2-1}$ whose kernel is residually torsion-free nilpotent.