Abstract : Right-angled Artin groups (RAAGs, for short) and their subgroups are of great interest because of their geometric, combinatorial and algorithmic properties. It is convenient to define these groups using finite simplicial graphs. The isomorphism type of the group is uniquely determined by the graph. Moreover, many structural properties of right angled Artin groups can be expressed in terms of their defining graph.
In this talk, I address the question of understanding the structure of a class of subgroups of RAAGs in terms of the graph. Bestvina and Brady, in their seminal work, studied these subgroups (now called Bestvina-Brady groups or Artin kernels) from finiteness conditions viewpoint. A Bestvina--Brady group is the kernel of the group homomorphism from a RAAG to $\ZZ$ sending all generators of the RAAG to 1. Unlike the RAAGs the isomorphism type of Bestvina-Brady groups is not uniquely determined by the defining graph. We prove that certain finitely presented Bestvina-Brady groups can be expressed as an iterated amalgamated product of RAAGs. Moreover, we show that this amalgamated product can be read off from the graph defining the ambient RAAG. In this talk, I also discuss the case where all the generators of RAAG are not mapped to 1. I give the presentation of the kernel of a map which maps the generators of a RAAG to some integer.