Cartesian subgroups in graph products of groups

Abstract

The kernel of the natural projection of a graph product of groups onto their direct product is called the Cartesian subgroup of the graph product. This construction generalises commutator subgroups of right-angled Coxeter and Artin groups. Using theory of polyhedral products, we give a lower and an upper bound on the number of relations in presentations of Cartesian groups and on their deficiency. The bounds are related to the fundamental groups of full subcomplexes in the clique complex, and the lower bound coincide with the upper bound if these fundamental groups are free or free abelian. Following Li Cai's approach, we also describe an algorithm that computes "small" presentations of Cartesian subgroups.