Combination of convergence groups

Abstract

We state and prove a combination theorem for relatively hyperbolic groups seen as geometrically finite convergence groups. For that, we explain how to contruct a boundary for a group that is an acylindrical amalgamation of relatively hyperbolic groups over a fully quasi-convex subgroup. We apply our result to Sela's theory on limit groups and prove their relative hyperbolicity. We also get a proof of the Howson property for limit groups.

Published in Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol7/paper27.abs.html

Cited by in corpus (48)

• Relative hyperbolicity and relative quasiconvexity for countable groups
• Limit groups for relatively hyperbolic groups, II: Makanin-Razborov diagrams
• Limit groups and groups acting freely on R^n-trees
• On the structural theory of II_1 factors of negatively curved groups, II: Actions by product groups
• The conjugacy problem for relatively hyperbolic groups
• Finiteness properties of cubulated groups
• On strongly quasiconvex subgroups
• Existential questions in (relatively) hyperbolic groups {\it and} Finding relative hyperbolic structures
• A Combination Theorem for Strong Relative Hyperbolicity
• Hyperbolic structures on groups
• Separation of Relatively Quasiconvex Subgroups
• The Isomorphism Problem for Toral Relatively Hyperbolic Groups
• Accidental parabolics and relatively hyperbolic groups
• Complexes of Groups and Boundaries
• Stallings graphs for quasi-convex subgroups
• A Combination Theorem for Convex Hyperbolic Manifolds, with Applications to Surfaces in 3-Manifolds
• OE and W* superrigidity results for actions by surface braid groups
• Limit groups for relatively hyperbolic groups, I: The basic tools
• Makanin-Razborov diagrams for limit groups
• Embedding relatively hyperbolic groups in products of trees
• One-ended subgroups of graphs of free groups with cyclic edge groups
• Finite index subgroups of fully residually free groups
• Aspherical manifolds, relative hyperbolicity, simplicial volume and assembly maps
• Relative hyperbolicity for automorphisms of free products
• Deciding Isomorphy using Dehn fillings, the splitting case
• Compactifications and algebraic completions of Limit groups
• Limit sets of relatively hyperbolic groups
• Local cut points and splittings of relatively hyperbolic groups
• Actions, length functions, and non-archemedian words
• Statistical hyperbolicity of relatively hyperbolic groups
• Residual properties of automorphism groups of (relatively) hyperbolic groups
• Relatively Hyperbolic Groups have Semistabile Fundamental Group at Infinity
• Coronae of relatively hyperbolic groups and coarse cohomologies
• Algorithms detecting stability and Morseness for finitely generated groups
• Property (QT) for 3-manifold groups
• Finitely generated simple sharply 2-transitive groups
• Height in Splitting of Relatively Hyperbolic Groups
• A Jorgensen-Thurston theorem for homomorphisms
• Quantifying the Residual Properties of Gamma-Limit Groups
• Permute and conjugate: the conjugacy problem in relatively hyperbolic groups
• Height, Graded Relative Hyperbolicity and Quasiconvexity {\it (with Corrigendum)}
• Relatively hyperbolic groups with strongly shortcut parabolics are strongly shortcut
• Combination Theorems in Groups, Geometry and Dynamics
• Quasiconvexity of virtual joins and separability of products in relatively hyperbolic groups
• The Hanna Neumann Conjecture for graphs of free groups with cyclic edge groups
• Bounded subgroups of relatively finitely presented groups
• On fully residually-$\mathcal{R}$ groups
• Parabolic groups acting on one-dimensional compact spaces