Subgroups of direct products of limit groups over coherent right-angled Artin groups

Abstract

During the last 40 years, a large body of work has been directed to study the connection betweenfiniteness properties of groups and their algebraic and algorithmic properties. One of the early results isdue to Baumslag and Roseblade, who showed that while finitely generated subgroups of the directproduct of two free groups are wild and untractable, the finitely presented ones have nice algebraic andalgorithmic properties. This work was widely extended during the years to the class of finitely presentedresidually free groups viewed as subgroups of direct products of limit groups.In this thesis we continue this study and show that the good behaviour of finitely presented subgroupsextends to the class of finitely presented residually Droms RAAGs. More precisely, we give a completecharacterisation of finitely presented residually Droms RAAGs and we obtain a number of consequencesrelated to decision problems, growth of homology and the Bieri-Neumann-Strebel-Renz invariants. Wealso study the subgroup structure of direct products of limit groups over Droms RAAGs depending ontheir finiteness properties. Finally, we initiate the study of finitely presented subgroups of direct productsof $2$-dimensional coherent RAAGs.