A combination theorem for combinatorially non-positively curved\n complexes of hyperbolic groups

2018/07/26 by Alexandre Martin, Martin, Alexandre, Damian Osajda +1
Mathematics · #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds #Group Theory (math.GR)

paper · pdf · doi:10.48550/arxiv.1807.10016

openalex publication_date 2018/07/26 · openalex created_date 2022/08/04 · openalex updated_date 2026/07/28

Abstract

We prove a combination theorem for hyperbolic groups, in the case of groups\nacting on complexes displaying combinatorial features reminiscent of\nnon-positive curvature. Such complexes include for instance weakly systolic\ncomplexes and C'(1/6) small cancellation polygonal complexes. Our proof\ninvolves constructing a potential Gromov boundary for the resulting groups and\nanalyzing the dynamics of the action on the boundary in order to use Bowditch's\ncharacterization of hyperbolicity. A key ingredient is the introduction of a\ncombinatorial property that implies a weak form of non-positive curvature, and\nwhich holds for large classes of complexes. As an application, we study the\nhyperbolicity of groups obtained by small cancellation over a graph of\nhyperbolic groups.\n

Related

• Complexes of groups and geometric small cancellation over graphs of groups
• A combinatorial non-positive curvature I: weak systolicity
• Graphical complexes of groups
• Hyperbolicity in non-metric cubical small-cancellation
• Combining relatively hyperbolic groups over a complex of groups