Splittings of triangle Artin groups

TL;DR

This paper investigates the structural decompositions of triangle Artin groups, demonstrating their splitting as amalgamated products or HNN extensions under certain conditions, and proves residual finiteness for all even three-generator cases.

Contribution

It provides new splitting results for triangle Artin groups and establishes residual finiteness for all even three-generator Artin groups.

Findings

Triangle Artin groups split as amalgamated products or HNN extensions under specified conditions.

All even three-generator Artin groups are residually finite.

Conditions for splitting depend on the parameters M, N, P in the group.

Abstract

We show that a triangle Artin group $\text{Art}_{MNP}$ where $M\leq N\leq P$ splits as an amalgamated product or an HNN extension of finite rank free groups, provided that either $M>2$, or $N>3$. We also prove that all even three generator Artin groups are residually finite.