# Structure and rigidity in hyperbolic groups I

@article{Rips1994StructureAR, title={Structure and rigidity in hyperbolic groups I}, author={Eliyahu Rips and Zlil Sela}, journal={Geometric \& Functional Analysis GAFA}, year={1994}, volume={4}, pages={337-371} }

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.

## 94 Citations

Homogeneity of torsion-free hyperbolic groups

- Mathematics
- 2019

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

- Mathematics
- 2009

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

- Mathematics
- 2004

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

- Mathematics
- 1997

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

- Mathematics
- 1997

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

- Mathematics
- 2005

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…

ON THE FAILURE OF THE CO-HOPF PROPERTY FOR SUBGROUPS OF WORD-HYPERBOLIC GROUPS

- Mathematics
- 1999

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

- Mathematics
- 2001

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

- Mathematics
- 2012

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

- Mathematics
- 2004

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…

## References

SHOWING 1-10 OF 16 REFERENCES

Stable actions of groups on real trees

- Mathematics
- 1995

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

- Mathematics
- 1980

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

- Mathematics
- 1983

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

- MathematicsMathematical 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…