A combination theorem for combinatorially non-positively curved complexes of hyperbolic groups. (9th May 2021)

• Record Type: Journal Article
• Authors: MARTIN, ALEXANDRE; OSAJDA, DAMIAN
• Is Part Of: Mathematical proceedings of the Cambridge Philosophical Society. Volume 170:Part 3(2021)
• Page Start: 445
• Page End: 477
• Publication Date: 2021-05-09
• Subjects: 20F67, -- 20F65; Mathematics -- Periodicals; 510.5
• DOI: 10.1017/S0305004119000446
• Languages: English
• ISSNs: 0305-0041

Abstract

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