Presentations for subgroups of Artin groups 

W. Dicks and I. J. Leary

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.  

Proc. Amer. Math. Soc. 127 (1999) 343-348.