A Tessellation and a Group Acting on Trees

M.J.Dunwoody

Introduction An orientation preserving simple closed curve c in a closed surface Σ (not a 2-sphere

or a projective plane) induces a decomposition of π_{1}(Σ) either as a free product with amalgamation, if c is separating, or as an HNN-group if Σ − c is connected. The decomposition in the separating case is non-trivial if and only if the curve c is essential, i.e. it does not bound a disc. A finite set of non-intersecting orientation preserving simple closed curves give a decomposition of π_{1}(Σ) as the fundamental group of a graph of groups. This is because the decompositions corresponding to different non-intersecting curves are compatible, i.e. after carrying out the decomposition induced by one curve, the decomposition corresponding to the other curve can be regarded as a decomposition of a vertex group in the first decomposition, and vice versa.

Of course there are examples of pairs of orientation preserving simple closed curves in Σ that do not give rise to compatible decompositions. This happens if and only if one curve cannot be isotoped to be disjoint from the other curve.

The theory of JSJ-decompositions of finitely presented groups was initiated by Rips and Sela [8]. They showed that a finitely presented group G has a decomposition (called a JSJ-decomposition) as the fundamental group of a graph of groups so that any splitting of G occurs in a natural way in this decomposition. In fact it means that incompatible splittings of G over infinite cyclic subgroups must occur in a vertex group of the decomposition which is a surface group or a free group (surface with boundary group).

This theory can be viewed as generalizing the theory of JSJ-decompositions of 3manifolds.

It was subsequently shown by Dunwoody and Sageev [4] and by Fujiwara and Papasoglu [5] that finitely presented groups have JSJ-decompositions which capture all splittings over slender subgroups. Such a splitting corresponds to a non-trivial action on a simplicial R-tree.

It was also shown by Bestvina and Feighn [1] that a finitely presented group has a decomposition similar to a JSJ-decomposition which captures a stable action of the group on an R-tree with small arc stabilizers.

This paper is devoted to a detailed examination of a particular group G, its actions on R-trees and its JSJ-decompositions.

It is shown that G has an action on an R-tree S which has infinite cyclic arc stabilizers. The action on S is geometric (see [7]) but not stable.

The example

Let G be the group with presentation

G = ⟨a, b, c, d, z|abcd = dcba, a^{−1}za = z^{2}, b^{−1}zb = z^{2}, c^{−1}zc = z^{2}, d^{−1}zd = z^{2}⟩.

Let N be the normal closure of z in G. So G is the splitting extension of N by H, the subgroup generated by {a, b, c, d} which is an orientable surface group of genus 2.

Let D = {n/2^{m}|n, m ∈ Z} be the dyadic rationals. There is an action of H on D, in which each of a, b, c, d maps to the automorphism which is multiplication by 2. Let G_{1} be the group which is the splitting extension of D by H, with this action of H. Clearly there is a surjective homomorphism from G to G_{1} which restricts to the identity map on H, and takes z to 1 ∈ D. We will show that this map is not an isomorphism and so N is not locally cyclic.

Consider a tessellation of the hyperbolic plane by octagons. Let Γ be the graph determining this tessellation. Thus Γ is an 8-regular plane graph, whose dual graph Γ′ is also 8-regular. There is an action of H on the hyperbolic plane by isometries which induces an action on Γ in which there is one orbit of vertices and 4 orbits of edges. Diametrically opposite edges of an octagonal face lie in the same edge orbit. Let A be one of the faces of Γ. Orient the edges of A as shown in Fig 1.

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a  a b  bc c  d d  Fig 1

Extend this orientation to an orientation O of EΓ in such a way that the orientation is preserved under the action of H. Define ϕ : V Γ → Z in the following way. For the left hand vertex of A put ϕ(v) = 0. For any other vertex u, choose an oriented edge path e_{1}, e_{2}, …, e_{n} from v to u. let ϕ(u) = ϵ_{1} + ϵ_{2} + … ϵ_{n} where ϵ_{i} = 1 if e_{i} is in O and ϵ_{i} = −1 if ē_{i} is in O. The fact that ϕ is well defined follows from the fact that it is the unique function such that ϕ(v) = 0 and such that δϕ(e) = 1 for each edge in O.

Let p, q ∈ V Γ. Consider a vertex path α = (v_{1}, v_{2}, …, v_{n}) from p to q, so that p = v_{1} and q = v_{n} and v_{i}, v_{i+1} are adjacent for i = 1, 2, …, n − 1. Define ν(α) = min{ϕ(v_{1}), ϕ(v_{2}), …, ϕ(v_{n})}, and put c(p, q) to be the maximum of ν(α) as α ranges over all vertex paths from p to q.

This maximum exists since ν(α) ≤ min(ϕ(p), ϕ(q)). Note that if p, q, r ∈ V Γ, we have

c(p, q) ≥ min{c(p, r), c(r, q)}.

This means that {c(p, q), c(p, r), c(q, r)} is an isosceles triple (see [2]), i.e. at least two of the values are equal and not greater than the third. If we put d(p, q) = ϕ(p) + ϕ(q) − 2c(p, q), then d is a pseudometric.Thus d(p, q) ≥ 0, and for example, if c(p, q) = c(r, p) ≤ c(r, q), then ϕ(r) ≥ c(r, q) and so ϕ(r) − c(r, p) ≥ c(r, q) − c(r, p) and d(p, r) + d(r, q) = ϕ(p) + ϕ(q) + 2ϕ(r) − 2c(p, r) − 2c(r, q) ≥ ϕ(p) + ϕ(q) + c(r, q) − c(r, p) − c(p, r) − c(r, q) ≥ ϕ(p) + ϕ(q) − 2c(p, q) = d(p, q). The other cases are similar.

If we identify all pairs of points p, q for which d(p, q) = 0 we get a metric space M (with integer valued distance function) which in fact is a tree. To see this, note that if d(p, q) = 0 then ϕ(p) = ϕ(q) = c(p, q). Thus ϕ takes a single value on each equivalence class of M and so defines an integer valued function - also denoted ϕ - on M. Let p, q ∈ M. A path in M is a set of points v_{1}, v_{2}, …, v_{n}, such that d(v_{i}, v_{i+1}) = 1 for i = 1, 2, …, n−1. It is a geodesic if n = d(v_{1}, v_{n}). Let p′, q′ be representives in V Γ of p, q. Given a path joining p, q in M there is a path in Γ joining p′, q′ for which the minimum value of ϕ attained on the path joining p′, q′ in Γ is the same as that joining p, q in M. It follows that we obtain a well defined value c(p, q) which is c(p′, q′) for any representatives p′, q′ of p, q respectively. Also for any path joining p, q the minimum value of ϕ(v_{i}) is at most c(p, q). A geodesic joining p, q has length ϕ(p)+ϕ(q)−2c(p, q). If the minimum value of ϕ(v_{i}) on the geodesic is m then the length of the geodesic is at least ϕ(p)−m+ϕ(q)−m. But m ≤ c(p, q). Hence m = c(p, q), and there is exactly one point in the geodesic where this value is attained. If q, q′ are adjacent to p in M and ϕ(q) = ϕ(q′) = ϕ(p) − 1, then c(p, q) = c(p, q′) = ϕ(q). It follows that d(q, q′) = 0 and so q = q′. The above analysis shows that for any geodesic there is a unique minimal value for ϕ(v_{i}) and the values of ϕ decrease monotonically from either end towards this minimal value. However this means that the path traversed is unique. Thus there is exactly one geodesic joining p, q. Let p, q, r ∈ M. We know that c(p, q), c(p, r), c(p, r) is an isosceles triple. If c(p, q) = c(q, r) ≤ c(p, r) then the unique point of [p, q] where ϕ takes the value c(p, q) must be the same as the unique point of [q, r] where ϕ takes this value. Either this point is q or the geodesics intersect in another point. Suppose this point is q, then it is not hard to see that the point on [q, r] where ϕ takes the value c(p, r) is the same as the point on [r, p] where ϕ takes the value c(p, r). Thus either this point is q and the union of [p, q] and [q, r] is [p, r] or the geodesics [p, q], [q, r] intersect in points other than q. It follows that M is a tree. We can direct the edges of M so that for any edge e, ϕ(ιe) < ϕ(τe). If we do this every vertex p has at most one edge directed into it, i.e. there is at most one edge for which τe = p. Construct a tree of groups on M as follows. For each point p ∈ M we have an infinite cyclic group generated by z_{p}. For each edge e there is an infinite cyclic group generated by z_{e}. We identify z_{e} = z_{τe}^{2}, z_{e} = z_{ιe}. Let N_{1} be the fundamental group of this tree of groups.

Consider a vertex u ∈ V Γ. We can extend ϕ linearly to EΓ and then to the hyperbolic plane, and we regard ϕ as defining a height function. The contours in A will be as in Fig 1 and in the neighbourhood of a vertex as in Fig 2. If we start at the centre of A there is a path (proceeding upwards) to the boundary intersecting only translates of edge a and another path intersecting only translates of edge b. The value of ϕ on either of these paths is monotone decreasing. Similarly there is a path proceeding downwards which only intersects translates of edge c (or of d) and on these paths also ϕ is monotone decreasing.

Now construct a group N_{2} as follows. For each v ∈ V Γ there is a copy of Z generated (multiplicatively) by z_{v}. For each oriented edge e with ιe = p and τe = q we have the relation z_{p}^{2} = z_{q}. In fact N_{1} and N_{2} are isomorphic. For suppose p, q ∈ V Γ and d(p, q) = 0, which means that ϕ(p) = ϕ(q) and there is a vertex path joining p and q for which every intermediate vertex v has ϕ(v) ≥ ϕ(p). Using induction on the number of local maxima of ϕ as one goes along the path, one proves that z_{p} = z_{q}.

As remarked above there is an action of H on the hyperbolic plane by isometries, which induces an action on Γ. If in Fig 2 the edges incident with u are labelled as shown then each of a, b, c, d takes u to an adjacent vertex by a hyperbolic isometry. This action now gives an action of H on N_{2} in which h^{−1}z_{u}h = z_{h(u)}. Thus a(u) = v, b(u) = w, c(u) = x, d(u) = y. It is not hard to see that N_{2} is isomorphic to N. If we put ϕ(u) = 0, then ϕ(v) = ϕ(w) = ϕ(x) = ϕ(y) = 1, and ϕ(v′) = ϕ(w′) = ϕ(x′) = ϕ(y′) = −1. Now ϕ(v) = ϕ(w) = 1 but v, w represent different points of M since c(v, w) = 0. To see this note that the shortest path from v to w passes through u and ϕ(u) = 0. There is no path from v to w for which ϕ only takes positive values. This is because on either side of the saddle point u between v and w there are paths to the boundary on which ϕ is monotone decreasing. On the other hand, c(v, y) = 1 as there is a path from v to y passing round 6 sides of an octagon for which ϕ takes positive values. A similar argument to that for v, w gives c(v, x) = c(x, w) = 0. Note this means that z_{v}, z_{x}, z_{w} are all distinct elements of N_{2} whose squares are equal to z_{u}. In particular this means that N is not locally cyclic.

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 Fig 2 v  w′  x  y′ y  x′  w  v′ • • •  •  • • •  •  •

Consider now a track in the hyperbolic plane. In this context, this means a connected 1-dimensional subset whose intersection with an octagonal region is either empty or it is a segment joining two points in the interior of distinct edges. In particular a track has empty intersection with V Γ. We will show that a track determines a decomposition of N as a free product with amalgamation.

Consider the complex of groups as illustrated in Fig 3. Thus X is the cell complex with 3 2-cells, 6 1-cells and one vertex. Then V (X) is the set of cells of X regarded as the vertex set of barycentric subdivision C(X) of X and E(X) is the set of edges of this subdivision. Thus V (X) has 10 elements. Note that the outside edges are identified in pairs as indicated.

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c d  d e  f  g 22 22 2  2 22 22  2 22 22  2 22 Fig 3

To define a complex of groups on C(X) we have to specify groups for each cell in V (X) and injective homomorphisms for each edge of E(X) so that the rules of [6] are satisfied. Note that the theory given in [6] is for a simplicial cell complex and some of our 2-cells have 4 faces rather than 3 faces. There is no real problem in doing this. It just means that one has to deal with fewer cells. For each v ∈ V (X), we assign an infinite cyclic group generated by z_{v}. The edges of E(X) correspond to pairs of cells (x, y) where the dimension of x is greater than that of y. The map corresponding to the pair (x, y) is an isomorphism z_{x} ↦ z_{y} except for the pairs marked with a 2 in Fig 3 for which the map corresponding to (x, y) is given by z_{x} ↦ z_{y}^{2}. The conditions of [6] Definition 2.1 are satisfied with g_{x,y} = 1 for all edges (x, y) ∈ E(X). A straightforward argument shows that G is the fundamental group of this complex of groups. This complex of groups is developable, i.e. there is a cell complex X̃ on which G acts so that X = G\X̃ and the complex of groups on C(X) is the one described above. This follows from [6] Theorem 4.1 if we can show that z generates an infinite cyclic subgroup of G. But there is a homomorphism of G in which z is mapped to an element of infinite order and so this is clear.

An essential simple closed curve in the pretzel P (orientable surface of genus two) determines a splitting of the pretzel group H. Thus H acts on the universal cover P̃. We can give P a hyperbolic structure, in which case P̃ is the hyperbolic plane and H acts by isometries. An essential simple closed curve in P, i.e. one which does not bound a disc, lifts to collection of curves in P̃ invariant under H. Since each such curve separates, the dual graph is an H-tree. By Bass-Serre theory this determines the splitting of H. This is described in detail for a general finitely presented group in [3], Chapter VI. It is also shown there that any action of a finitely presented group G on a simplicial tree is resolved by such a geometric splitting, i.e. if S is a simplicial G-tree then there os a geometric simplicial G-tree T for which there is a G-morphism θ : T → S. If p ∈ T then the stabilizer G_{p} is a subgroup of G_{θ(p)}. Thus if there is a splitting of G over a slender (small) subgroup then there must be a geometric splitting over a slender (small) subgroup. In our case instead of the pretzel group H we have the fundamental group G of a complex of groups corresponding to a complex which is a cell decomposition of a pretzel. The same theory will work however in this case, if we can show that in any action of G on a simplicial tree each cell group, i.e. a group corresponding to a vertex of C(X), fixes a vertex of the simplicial tree. Thus we need to show that in any action of G on a simplicial tree z does not act hyperbolically.

Suppose z acted hyperbolically with axis A_{z}. Now any two elements of N have a power in common. It follows from [2] Chapter 3, Lemma 1.7, that all elements of N have a common axis A_{z}. Also A_{gzg^{−1}} = gA_{z} and so A_{z} = gA_{z} for every g ∈ G and we have an abelian action. But z is in the derived subgroup of G and so we have a contradiction.

The cell complex C(X) lifts to a cell complex C(Y) based on the tessellation of the hyperbolic plane described above. We know that G is the fundamental group of the complex of groups specified. Assigning groups to C(Y) so that the indices in each translate of the fundamental region is as in Fig 3, we obtain a complex of groups whose fundamental group is N. Suppose we have a track t in Y. Now t separates Y into two pieces L, R. We assume that L, R are closed regions so that L ∩ R = t. We show that t induces a decomposition of N as a free product with amalgamation. Let Y_{1} be a lift of Y to X̃. Then N is generated by the stabilizers of vertices of Y_{1}.

The tracks we are interested in are non-compact and intersect each 2-cell of Y at most once. For most of the tracks we consider the track can be oriented so that in each translate of the fundamental region the arrows go from top to bottom. Thus for example a component of the lift of the simple closed curve shown in Fig 4(a) is such a track, but not the simple closed curve of Fig 4(b) .

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_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} Fig 4(a)  Fig 4(b)

Let N_{R}, N_{t} be the subgroup of N generated by vertices of Y_{1} corresponding to vertices of R, L repectively, and let N_{t} = N_{R} ∩ N_{L}. In fact N_{t} is generated by the stabilizers of 1-cells of Y_{1} which intersect the lift of t. Then N = N_{R} ∗_{N_{t}} N_{L}. To see this, note that we have shown that N is the fundamental group of a tree of cyclic groups, the vertices of which are equivalence classes of vertices of Γ. Adjacent vertices in Γ which are not in the same equivalence class are adjacent in this tree. The group N_{R} will be generated by the groups corresponding to the vertices of Γ which lie in R, which is connected. Thus N_{R} is the subgroup of N which is the fundamental group of a connected subgraph, i.e. a subtree. The same must be true for L and, hence, also for t, since the intersection of two subtrees is a subtree.

Let us consider some examples. The first example shown in Fig 4(a) lifts to a track which has a small stabilizer. The stabilizer in N is locally cyclic. It induces a splitting of G as an HNN-group over a small subgroup which is a locally cyclic group extended by a cyclic group. Any track, as described above gives a two way infinite sequence of ϕ values corresponding to the ϕ value of successive 1-cells that it intersects. The sequence for the first example goes …, 0, 1, 1, 1, 0, 1, 1, 2, 2, 2, … and satisfies the property that the n+7-th term is the n-th term plus one. Note that although the sequence is not monotone, the corresponding cell stabilizers are totally ordered by inclusion. In the second example, shown in Fig 5 the track has a non small stabilizer. There is a lifting track on which ϕ takes only values 0, 1, 2. The sequence, which is periodic with period 8 is …, 1, 2, 2, 2, 1, 1, 0, 1, 2, 2, . . .. In this case the groups of the cells it intersects are not totally ordered by inclusion.

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1  1 2 3  3 4 5  5 6 7  7 8^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}  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The group G does have non-trivial splittings over slender subgroups. If fact it has a JSJ-decomposition in which there are two compatible splittings corresponding to the curves shown in Fig 6.

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Another interesting property of G is that it is non-Hopf. Thus there is a homomorphism ν : G → G in which ν(a) = a, ν(b) = b, ν(c) = c, ν(d) = d, ν(z) = z^{2}. This homomorphism is surjetive, but not injective, since if u, v are two vertices of Y for which ϕ(u) = ϕ(v) = c(u, v) + 1 then z_{u} ≠ z_{v} but ν(z_{u}) = ν(v).

Also G admits actions on R-trees in which point stabilizers are locally cyclic and arc stabilizers are cyclic.

We now generalize the example of Stallings [9]. We choose positive real numbers α, β, γ, δ. Draw our octagon A so that the projections on the x-axis of the sides have lengths α, β etc. Foliate A by vertical lines. This foliation lifts to a foliation both of Y and of X̃. The leaves of the foliation on Y can be regarded as the points of an R-tree. The action of H on this tree is free if the values α, β, γ, δ are rationally independent. We show that the set S of leaves of X̃ can also be regarded as the points of an R-tree. There is a transverse measure on X̃, obtained by lifting the dx measure on X which also lifts the measure on Y. We get a pseudometric on S by taking the infimum of integrals of the measure along smooth curves joining points in the leaves. We need to show that the distance between distinct leaves is never zero. If two leaves in X̃ project to different leaves in Y then the distance between them is at least as big as the distance in Y which is non zero. Suppose then that we have two leaves which project to the same leaf in Y. If this leaf does not contain a vertex then it is a track t and we have seen that a track determines a decomposition of N as a free product with amalgamation and the lifts of t in X̃ can be regarded as the edges of a simplicial N-tree with one orbit of edges. Clearly it suffices to show that tracks which are adjacent edges in this simplicial tree are a non-zero distance apart. Two such tracks are t̃ and nt̃ where n ∈ N_{R} or n ∈ N_{L}. If say n ∈ N_{R}, then there there is a finite set F of vertices of Y_{R} such that n is in the subgroups generated by z_{f}, f ∈ F. We can find a track t_{1} in the foliation of Y which separates F from t. The lifts of t and t_{1} to X̃ will give a simplicial N-tree in which t̃and nt̃are no longer adjacent - they will be separated by a lift of t_{1}. But we have seen that a lift of t is always non-zero distance from a lift of t_{1}, and so the argument is complete in this case. If the leaf contains a vertex, it projects to a leaf in Y which contains a vertex. If the values α, β, γ, δ are rationally independent then the leaf in Y will contain just one vertex and it will divide the plane up into finitely many regions. There will be a decompositon of N into a tree of groups with one vertex group for the leaf and a vertex for each region. An argument similar to the one above then shows that any two lifts of the leaf to X̃ are a non-zero distance apart.

It follows then that S is an R-tree. We now show that we can choose the values α, β, γ, δ so that in the action of G on S point stabilizers are locally cyclic and arc stabilizers are cyclic. What we want to arrange is that the sequence of values of ϕ on each track of the foliation of Y is monotone. This is the case if each of the three vertices in the top half of the polygon A is to right of the corresponding vertex in the bottom half of the polygon, i.e. if α > δ, α+β > δ +γ. It is clear that α, β, γ, δ can be chosen to be rationally independent and to satisfy these inequalities. An arc stabilizer must be contained in the intersection of stabilizers of two distinct points and it is not hard to see that this must be cyclic, as the corresponding tracks will eventually intersect distinct sets of 1-simplices.

Concluding remarks Let C_{n} be the class of groups that are virtually polycyclic with a fixed Hirsch number

n, and let ZLC_{n} be the class of groups that are extensions of a group which is locally C_{n} by an infinite cyclic group. I think that there may be a JSJ-theory for splittings of a finitely presented group G over ZLC_{n}-subgroups in which there are vertex groups that are like those described in the example. It may also be the case that any action of G on an R-tree with arc stabilizers in C_{n} can be read off from such a JSJ-decomposition.

References

[1] M.Bestvina and M.Feighn, Stable actions of groups on real trees, Invent. Math. 121 (1995), 287 -321.

[2] I.M.Chiswell, Introduction to Λ-trees, World Scientific, 2001.

[3] W.Dicks and M.J.Dunwoody, Groups acting on graphs, Cambridge University Press, 1989.

[4] M.J.Dunwoody and M.E.Sageev, JSJ-Splittings for finitely presented groups over slender subgroups, Invent. Math. 135 (1999) 25-54.

[5] K.Fujiwara and P.Papasoglu, JSJ-Decompositions of finitely presented groups and complexes of groups, preprint (1998).

[6] A.Haefliger, Complexes of groups and orbihedra, in Group theory from a geometric viewpoint, Eds. Ghys,Haefliger, Verjovsky, World Scientific 1991.

[7] G.Levitt and F.Paulin, Geometric group actions on trees, Amer. J. Math 119 (1997) 83-102.

[8] E.Rips and Z.Sela, Cyclic splittings of finitely presented groups and the canonical JSJdecomposition, Annals of Maths. 146 (1997) 53-104.

[9] J.R.Stallings, Adian groups and pregroups, in Essays in group theory, MSRI publications 8 (edited by S.M.Gersten) (1987) 321-342.

Department of Mathematics University of Southampton Southampton SO17 1BJ M.J.Dunwoody@maths.soton.ac.uk