Groups of cohomological codimension one

Key Points

• The aim is to explore the structural properties of commensurated subgroups and their relationship to cohomological dimensions.
• Investigating the properties of commensurated subgroups in groups of type VFP.
• Analyzing the relationship between the virtual cohomological dimensions of groups H and G.
• Studying specific cases involving one-relator groups and duality groups.
• Demonstrated that if group G has a commensurated subgroup H, then G acts as the fundamental group of a graph of groups.
• Showed that all vertex and edge groups in this structure are commensurable to H.
• Explored the behavior of commensurated subgroups in the context of one-relator groups.