Structure and rigidity in hyperbolic groups I

  title={Structure and rigidity in hyperbolic groups I},
  author={Eliyahu Rips and Zlil Sela},
  journal={Geometric \& Functional Analysis GAFA},
  • E. Rips, Z. Sela
  • Published 1 May 1994
  • Mathematics
  • Geometric & Functional Analysis GAFA
We introduce certain classes of hyperbolic groups according to their possible actions on real trees. Using these classes and results from the theory of (small) group actions on real trees, we study the structure of hyperbolic groups and their automorphism group. 
Homogeneity of torsion-free hyperbolic groups
We give a complete characterization of torsion-free hyperbolic groups which are homogeneous in the sense of first-order logic, in terms of the JSJ decompositions of their free factors.
Elementary embeddings in torsion-free hyperbolic groups
We consider embeddings in a torsion-free hyperbolic group which are elementary in the sense of first-order logic. We give a description of these embeddings in terms of Sela's hyperbolic towers. We
The Hopf Property for Subgroups of Hyperbolic Groups
A group is said to be Hopfian if every surjective endomorphism of the group is injective. We show that finitely generated subgroups of torsion-free hyperbolic groups are Hopfian. Our proof
Symmetric patterns of geodesics and automorphisms of surface groups
Abstract. We prove a non-equivariant version of Mostow rigidity for symmetric patterns of geodesics in hyperbolic space. This result allows for a classification of pseudo-Anosov surface group
Cyclic Splittings of Finitely Presented Groups and the Canonical JSJ-Decomposition
The classification of stable actions of finitely presented groups on ℝ-trees has found a number of applications. Perhaps one of the most striking of these applications is the theory of canonical
Limit groups for relatively hyperbolic groups, II: Makanin-Razborov diagrams
Let be a torsion-free group which is hyperbolic relative to a collection of free abelian subgroups. We construct Makanin–Razborov diagrams for . We also prove that every system of equations over is
We provide an example of a finitely generated subgroupH of a torsion-free word-hyperbolic group G such that H is one-ended, and H does not split over a cyclic group, and H is isomorphic to one of its
On the failure of the co-hopf property for subgroups of word-hyperbolic groups
We provide an example of a finitely generated subgroupH of a torsion-free word-hyperbolic groupG such thatH is one-ended, andH does not split over a cyclic group, andH is isomorphic to one of its
Test elements in torsion-free hyperbolic groups
We prove that in a torsion-free hyperbolic group, an element is a test element if and only if it is not contained in a proper retract.
Height in splittings of hyperbolic groups
SupposeH is a hyperbolic subgroup of a hyperbolic groupG. Assume there existsn > 0 such that the intersection ofn essentially distinct conjugates ofH is always finite. Further assumeG splits overH


Stable actions of groups on real trees
This paper further develops Rips's work on real trees. We study a class of actions called ‘stable’ which includes actions with trivial arc stabilizers and small actions of hyperbolic groups.
Surfaces and Planar Discontinuous Groups
Free groups and graphs.- 2-Dimensional complexes and combinatorial presentations of groups.- Surfaces.- Planar discontinuous groups.- Automorphisms of planar groups.- On the complex analytic theory
A simple presentation for the mapping class group of an orientable surface
LetFn.k be an orientable compact surface of genusn withk boundary components. For a suitable choice of 2n + 1 simple closed curves onFn,1 the corresponding Dehn twists generate bothMn,o andMn,1. A
A finite set of generators for the homeotopy group of a 2-manifold
  • W. Lickorish
  • Mathematics
    Mathematical Proceedings of the Cambridge Philosophical Society
  • 1964
The homeotopy group Λx of a space X is the group of all homeomorphisms of X to itself, modulo the subgroup of those homeomorphisms that are isotopic to the identity. In this paper X will be taken to