No nontrivial baumslag-solitar relations in extended admissible groups

Tóm tắt

In this work, we analyze the structural attributes of extended admissible groups, which constitute a diverse family of finitely generated groups. These groups serve as a generalization for both the graph-of-groups decompositions linked to non-geometric 3-manifolds and the standard admissible groups introduced by Croke and Kleiner. Our principal finding demonstrates that extended admissible groups do not support any nontrivial Baumslag–Solitar relations. This result generalizes a classical theorem establishing that the fundamental groups of 3-manifolds are free from such relations. By blending geometric group theory methodologies with Bass–Serre theory, our proof examines the group action on the corresponding Bass–Serre tree alongside the behavior of relatively hyperbolic vertex groups. This conclusion enhances our understanding of rigidity phenomena within relatively hyperbolic settings and offers additional validation for the quasi-isometric rigidity of extended admissible groups.