Recognizing a relatively hyperbolic group by its Dehn fillings

2015/06/30 by François Dahmani, Vincent Guirardel · 16 citations
Computer Science · Mathematics · #3-manifold #Dehn surgery #Fundamental group #Geometric and Algebraic Topology #Group (periodic table) #Homotopy and Cohomology in Algebraic Topology #Hyperbolic 3-manifold #Hyperbolic function #Hyperbolic group #Hyperbolic manifold #Hyperbolic triangle #Isomorphism (crystallography) #Knot (papermaking) #Mathematical analysis #Mathematics #Pure mathematics #Relatively hyperbolic group #Rigidity (electromagnetism) #Structural engineering #math.GR #math.GT #semigroups and automata theory

paper · pdf · doi:10.1215/00127094-2018-0014

published in Duke Mathematical Journal 167(12) (Duke University Press) · Minor modification (including typesetting). 56 pages

arxiv created 2018/04/18 · openalex publication_date 2018/07/20 · arxiv updated 2018/11/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

Dehn fillings for relatively hyperbolic groups generalize the topological Dehn surgery on a noncompact hyperbolic 3-manifold such as hyperbolic knot complements. We prove a rigidity result saying that if two nonelementary relatively hyperbolic groups without certain splittings have sufficiently many isomorphic Dehn fillings, then these groups are in fact isomorphic. Our main application is a solution to the isomorphism problem in the class of nonelementary relatively hyperbolic groups with residually finite parabolic groups and with no suitable splittings.

Citations

• Géométrie et théorie des groupes
• Negatively Oriented Ideal Triangulations and a Proof of Thurston's Hyperbolic Dehn Filling Theorem
• Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces
• The isomorphism problem for all hyperbolic groups
• The Isomorphism Problem for Hyperbolic Groups I
• Trees of cylinders and canonical splittings
• Three dimensional manifolds, Kleinian groups and hyperbolic geometry
• The isomorphism problem for toral relatively hyperbolic groups
• A topological characterisation of relatively hyperbolic groups
• Relatively Hyperbolic Groups
• Peripheral fillings of relatively hyperbolic groups
• Dehn filling in relatively hyperbolic groups
• Metric Spaces of Non-Positive Curvature
• RELATIVELY HYPERBOLIC GROUPS