Presentations for subgroups of Artin groups

TL;DR

This paper provides explicit presentations for certain subgroups of Artin groups, linking their algebraic structure to the topology of associated clique complexes, including finite presentations under specific topological conditions.

Contribution

It introduces a new presentation for the subgroup H(L) of Artin groups, connecting algebraic generators and relators to the topology of clique complexes, with finite presentations in simply-connected cases.

Findings

Presented a new generator and relator set for H(L)

Established finite presentations when the clique complex is simply-connected

Connected algebraic properties of H(L) to topological features of L

Abstract

For a connected graph L, let G(L) be a group with generators the vertex set of L, subject only to the relations that the ends of each edge commute. Now let H(L) be the kernel of the homomorphism from G(L) to the integers that takes each vertex to 1. M. Bestvina and N. Brady have shown that finiteness properties of H(L) are intimately related to the topology of the clique complex of L. We give a presentation for H(L), with generators the edges of L, and an infinite family of relators for each 1-cycle in L. In the case when the clique complex for L is simply-connected, we give a finite presentation for H(L), with generators the edges (or 2-cliques) of L, and two relators for each 3-clique in L.