Cartesian subgroups in graph products of groups

TL;DR

This paper studies the structure of Cartesian subgroups in graph products of groups, providing bounds on their presentations and an algorithm for computing simplified presentations, generalizing known subgroup concepts.

Contribution

It introduces bounds on the relations and deficiency of Cartesian subgroups and offers an algorithm for their simplified presentation, extending previous subgroup theories.

Findings

Bounds on the number of relations in Cartesian subgroups

Relation between bounds and fundamental groups of subcomplexes

Algorithm for computing small presentations of Cartesian subgroups

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.