A gluing theorem for negatively curved complexes

2015/10/31 by Samuel Brown · 2 citations
Computer Science · Mathematics · #Geometric and Algebraic Topology #Graph #Homotopy and Cohomology in Algebraic Topology #Hyperbolic manifold #Limit (mathematics) #Simplicial approximation theorem #Simplicial complex #Topological and Geometric Data Analysis #Vertex (graph theory) #math.GR #math.GT #msc:20F65 #msc:20F67

paper · pdf · doi:10.1112/jlms/jdw021

published in Journal of the London Mathematical Society 93(3), 741-762 (Wiley) · 26 pages. Typo corrected from previous version

openalex publication_date 2016/04/27 · openalex created_date 2016/06/24 · arxiv created 2016/07/09 · arxiv updated 2017/05/17 · openalex updated_date 2026/08/05

Abstract

A simplicial complex is called negatively curved if all its simplices are isometric to simplices in hyperbolic space, and it satisfies Gromov's Link Condition. We prove that, subject to certain conditions, a compact graph of spaces whose vertex spaces are negatively curved 2-complexes, and whose edge spaces are points or circles, is negatively curved. As a consequence, we deduce that certain groups are CAT ( - 1 ) . These include hyperbolic limit groups, and hyperbolic groups whose JSJ components are fundamental groups of negatively curved 2-complexes, for example, finite graphs of free groups with cyclic edge groups.

Citations

• Diophantine geometry over groups I : Makanin-Razborov diagrams
• The accessibility of finitely presented groups
• Trees of cylinders and canonical splittings
• Cut points and canonical splittings of hyperbolic groups
• Metric Spaces of Non-Positive Curvature
• Cyclic Splittings of Finitely Presented Groups and the Canonical JSJ Decomposition
• Hyperbolic Groups