Presentations of Subgroups of Artin Groups

Abstract

Let $A \Gamma$ be the Artin group based on the graph $\Gamma$, and let $\phi \colon A \Gamma \to {\mathbb Z}$ be a homomorphism which maps each of the standard generators of $A \Gamma$ to 0 or 1. We compute an explicit presentation for $\ker \phi$ in the general case. In the case where $\Gamma$ is a tree with a connected and dominating live subgraph, we prove $\ker \phi$ is a free group and we calculate its rank. In addition, if $A \Gamma$ is a 2-cone with live apex, we prove $\ker \phi$ is isomorphic to the Artin group on the base of the cone, and if $\Gamma$ is a special tree-triangle combination, we determine conditions on $\Gamma$ which ensure the finite presentation of $\ker \phi$.