# JSJ-splittings for finitely presented groups over slender groups

@article{Dunwoody1999JSJsplittingsFF, title={JSJ-splittings for finitely presented groups over slender groups}, author={M. J. Dunwoody and Michah Sageev}, journal={Inventiones mathematicae}, year={1999}, volume={135}, pages={25-44} }

Abstract. We generalize the JSJ-splitting of Rips and Sela to give decompositions of finitely presented groups which capture splittings over certain classes of small subgroups. Such classes include the class of all 2-ended groups and the class of all virtually Z⊕Z groups. The approach, called “track zipping”, is relatively elementary, and differs from the Rips-Sela approach in that it does not rely on the theory of R-trees but rather on an understanding of certain embedded 1-complexes (called…

## 117 Citations

JSJ-Decompositions of finitely presented groups and complexes of groups

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AbstractWe give an example of two JSJ decompositions of a group that are not
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Abstract We construct a ‘structure invariant’ of a one-ended, finitely presented group that describes the way in which the factors of its JSJ decomposition over two-ended subgroups fit together. For…

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A hierarchy of a group is a rooted tree of groups obtained by iteratively passing to vertex groups of graphs of groups decompositions. We define a (relative) slender JSJ hierarchy for (almost)…

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- 2021

The celebrated Stallings’ decomposition theorem states that the splitting of a finite index subgroup H of a finitely generated group G as an amalgamated free product or an HNN-extension over a finite…

JSJ-Decompositions of Coxeter Groups over Virtually Abelian Splittings

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- 2008

The idea of “JSJ-decompositions” for 3-manifolds began with work of Waldhausen and was developed later through work of Jaco, Shalen and Johansen. It was shown that there is a finite collection of…

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- Mathematics
- 2001

We describe a means of constructing splittings of a one-ended finitely generated group over two-ended subgroups, starting with a finite collection of codimension-one two-ended subgroups. In the case…

JSJ-Decompositions of Coxeter Groups over Virtually

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- 2008

The idea of “JSJ-decompositions” for 3-manifolds began with work of Waldhausen and was developed later through work of Jaco, Shalen and Johansen. It was shown that there is a finite collection of…

Boundaries and JSJ Decompositions of CAT(0)-Groups

- Mathematics
- 2007

Let G be a one-ended group acting discretely and co-compactly on a CAT(0) space X. We show that ∂X has no cut points and that one can detect splittings of G over two-ended groups and recover its JSJ…

## References

SHOWING 1-10 OF 21 REFERENCES

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…

Structure and Rigidity in (Gromov) Hyperbolic Groups and Discrete Groups in Rank 1 Lie Groups II

- Mathematics
- 1997

Abstract. We borrow the Jaco-Shalen-Johannson notion of characteristic sub-manifold from 3-dimensional topology to study cyclic splittings of torsion-free (Gromov) hyperbolic groups and finitely…

Cut points and canonical splittings of hyperbolic groups

- Mathematics
- 1998

In this paper, we give a construction of the JSJ splitting of a one-ended hyperbolic group (in the sense of Gromov [Gr]), using the local cut point structure of the boundary. In particular, this…

An analogue of the torus decomposition theorem for certain Poincaré duality groups

- Mathematics
- 1990

It is shown that Poincare duality groups which satisfy the maximal condition on centralisers have a canonical decomposition as the fundamental group of a finite graph of groups in which the edge…

Seifert fibered spaces in 3-manifolds

- Mathematics
- 1979

Publisher Summary This chapter describes Seifert Fibered Spaces in 3-Manifolds. There exist finitely many disjoint, non-contractible, pairwise non-parallel, embedded 2-spheres in M, whose homotopy…

Bounding the complexity of simplicial group actions on trees

- Mathematics
- 1991

We shall state the main result of this paper in terms of group actions on simplicial trees. Suppose that a group G acts simplicially on a tree T without inversions. For brevity we say that Tis a…

Cutting up graphs

- MathematicsComb.
- 1982

It is shown that there is a subsetd ⊂V Γ which has the following properties: bothd andd*=VΓ\d are infinite.

Groups acting on graphs

- Mathematics
- 1991

In the first part of this thesis we investigate the automorphism groups of regular trees. In the second part we look at the action of the automorphism group of a locally finite graph on the ends of…

A group theoretic criterion for property FA

- Mathematics
- 1996

We give group-theoretic conditions on a set of generators of a group G which imply that G admits no non-trivial action on a tree. The criterion applies to several interesting classes of groups,…

Some examples of hyperbolic groups

- Mathematics
- 1999

We describe some examples of hyperbolic groups to compare the range of validity of diierent theories of JSJ-decompositions of groups.