Combination of convergence groups

By Franccois Dahmani. 2002

279 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e63c63a3190ab6531bb9d88380c5034fb9ce2a2e

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Asymptotic dimension of relatively hyperbolic groups (2004, background): “The class of relatively hyperbolic groups includes many examples of interest such as fundamental groups of finite-volume Riemannian manifolds of pinched negative curvature [4, 11], geometrically finite convergence groups [18], small cancellation quotients of free products [16], fully residually free”
Relatively hyperbolic groups: Intrinsic geometry, algebraic properties, and algorithmic problems (2004, result): “morphic to the boundary of G. In the case of ordinary hyperbolic groups (i.e. in case the set of bounded parabolic points of Mis empty) this result was obtained by Bowditch [11]. (V) Recently Dahmani [32] proved combination theorem for relatively hyperbolic groups. This allowed to show that the limi”
The outer space of a free product (2005, background): “This applies to groups acting freely on R-trees, which are hyperbolic relative to their maximal abelian subgroups [9].”
Some geometric groups with Rapid Decay (2003, background): “More on relatively hyperbolic groups can be found in [ D ] or [DrS1].”
Relative hyperbolicity and relative quasiconvexity for countable groups (2008, background): “Two of these formulations were Published: 3 September 2010 DOI: 10.2140/agt.2010.10.1807 introduced by Dahmani [11] and Osin [27] in special cases.”
Limit groups for relatively hyperbolic groups. I. The basic tools (2004, unknown): “See [19, 6, 12, 44] for details.”
Dehn filling in relatively hyperbolic groups (2006, background): “(1) Hyperbolic groups are hyperbolic relative to the empty collection of subgroups; (2) Fundamental groups of geometrically finite hyperbolic manifolds are hyperbolic relative to the cusp subgroups; (3) Free products are hyperbolic relative to the free factors; (4) A group which acts properly and co”
Relatively hyperbolic groups: geometry and quasi-isometric invariance (2006, background): “f tori and Klein bottles not contained in a graphmanifold component; 5. fully residually free groups, also known as limit groups, are hyperbolic relative to their maximal Abelian non-cyclic subgroups [Dah03a]; 6. more generally, ﬁnitely generated groups acting freely on Rn–trees are hyperbolic relat”
Effective JSJ Decompositions (2004, background): “11 also follows from [12], because finitely generated fully residually free groups are relatively hyperbolic [20].”
RESEARCH ANNOUNCEMENT: THE STRUCTURE OF GROUPS WITH A QUASICONVEX HIERARCHY (2009, background): “This hierarchy allows one to prove that limit groups are hyperbolic relative to free abelian subgroups [11, 3].”
Hall’s Theorem for Limit Groups (2006, background): “Dahmani proved in [6] the stronger result that finitely generated subgroups of limit groups are quasi-convex in the sense of convergence groups.”
Normal automorphisms of relatively hyperbolic groups (2008, methodology): “The first author was supported by the Swiss National Science Foundation grant PP002-116899.”