Residual finiteness of certain 2-dimensional Artin groups

TL;DR

This paper proves that many 2-dimensional Artin groups are residually finite, including most 3-generator cases with certain labels, by demonstrating their decomposition into free products with amalgamation or HNN extensions of free groups.

Contribution

It establishes residual finiteness for a broad class of 2-dimensional Artin groups, introducing new splitting techniques and extending known results.

Findings

Many 2-dimensional Artin groups are residually finite.

Certain Artin groups split as free products with amalgamation or HNN extensions.

Residual finiteness holds for large type Artin groups with directed cycle graphs.

Abstract

We show that many $2$-dimensional Artin groups are residually finite. This includes $3$-generator Artin groups with labels $\geq 4$ except for $(2m+1, 4,4)$ for any $m\geq 2$. As a first step towards residual finiteness we show that these Artin groups, and many more, split as free products with amalgamation or HNN extensions of finite rank free groups. Among others, this holds for all large type Artin groups with defining graph admitting an orientation, where each simple cycle is directed.