Geometriae Dedicata 61: 29--41, 1996. 29 © 1996 KluwerAcademic Publishers. Printed in the Netherlands.

When is the Graph Product of Hyperbolic Groups Hyperbolic?

JOHN MEIER Department of Mathematics, Lafayette College, Easton, PA 18042, U.S.A. e-mail: meierj@lafayette.edu

(Received: 7 December 1994; accepted in final form: 23 November 1995)

Abstract. Given a finite simplicial graph G, and an assignment of groups to the vertices of if, the graph product is the free product of the vertex groups modulo relations implying that adjacent vertex groups commute. We use Gromov's link criteria for cubical complexes and techniques of Davis and Moussang to study the curvature of graph products of groups. By constructing a CAT(-1) cubical complex, it is shown that the graph product of word hyperbolic groups is itself word hyperbolic if and only if the full subgraph ~a in G, generated by vertices whose associated groups are finite, satisfies three specific criteria. The construction shows that arbitrary graph products of finite groups are Bridson groups.

Mathematics Subject Classifications (1991): 20F32, 53C23.

Key words: word hyperbolic groups, Bridson groups, cubical complexes, graph products.

1. Introduct ion

In a simplicial graph, two vertices are adjacent if they are joined by a single edge. Given a finite simplicial graph ~ and for each vertex v a group Gv, the graph product, denoted G~, is the free product of the vertex groups with added relations that imply elements of adjacent vertex groups commute.

Graph products have been actively studied by mathematicians and computer scientists (see [2], [14] and the references cited there). If all the vertex groups are infinite cyclic, then the graph product is referred to as a 'graph group' [8] and when the vertex groups are of order two, then the graph product is a 'right-angled Coxeter group' [5].

Graph products are known to have many nice closure properties. The graph product of residually finite groups is residually finite [11] and many geometric properties of groups are preserved by graph products: the graph product of semi- hyperbolic groups is semihyperbolic and the graph product of automatic groups is automatic [14] (see [1 ] for background on semihyperbolic groups, [9] for automatic groups).

Word hyperbolic groups are certainly one of the most actively studied classes of infinite groups. It is known that every infinite word hyperbolic group contains an infinite cyclic subgroup and that word hyperbolic groups do not contain copies

30 JOHN MEIER

of Z × Z. It follows that an arbitrary graph product of word hyperbolic groups cannot be word hyperbolic, because the construction of a graph product involves taking direct sums; the graph product of infinite word hyperbolic groups is not word hyperbolic, unless of course the underlying graph is a null graph. (For background on word hyperbolic groups, see [10].)

Because every finite group is word hyperbolic, there is still a window of possibil- ity open. There are nontrivial graph products of groups which are word hyperbolic.

DEFINITION. For each vertex v of ~, the link graph of v is the subgraph of generated by the vertices of G adjacent to v. We denote this g raph /~ .

Given any full subgraph ~ C ~, by [11] the graph product GT-/injects as a subgroup into GG. In particular, if G~ is the group associated to a vertex v, then the group G~ × G/~v is a subgroup of G~. Further, if Z is the subgraph formed by removing v from G, then E. Green shows that the graph product decomposes into a free product with amalgamation:

= × G£,v) *cc G Z .

It follows that, for any graph product of word hyperbolic groups, if there is a vertex v whose associated vertex group is infinite, and G£v is an infinite group, then the graph product contains a copy of Z × Z. Notice that GE~ is an infinite group unless all the vertex groups o f / ~ are finite and E~ is a complete graph. Thus, if a graph product of word hyperbolic groups is to be word hyperbolic, then it is necessary that no two vertices associated to infinite word hyperbolic groups are adjacent, and if v is a vertex associated to an infinite word hyperbolic group, £v is a complete graph.

Even if there are no vertex groups of infinite order, there is yet another way the graph product can contain a copy of Z × Z. If a full subgraph of ~ is a circuit of four edges, with vertex groups G1 , . . . , G4 listed in order, then the subgroup generated by this subgraph is the direct sum of the free products of opposite vertex groups, (G1 ~r G3) x (G2 ~k G4). Because the free products in this direct sum are virtually free, this subgroup contains at least one copy of Z × Z.

The main result of this paper is that the three obstructions outlined above are the only obstructions to a graph product of word hyperbolic groups being word hyperbolic.

MAIN THEOREM. Let G be a finite simplicial graph with word hyperbolic groups assigned to its vertices. Let ~ be the full subgraph o f f generated by the vertices associated with finite groups. The graph product GG is word hyperbolic if and only

if..

(i) The full subgraph generated by the vertices in ~ - ~ is a null graph; (ii) If v E ~ - ~F~, then £v is a complete graph;

WHEN IS THE GRAPH PRODUCT OF HYPERBOLIC GROUPS HYPERBOLIC? 3 1

(iii) Every circuit in 3r# of length four contains a chord.

The important step in establishing the Main Theorem is constructing a cubical complex admitting a discrete, cocompact action of G.T¢. The same basic construc- tion applies to arbitrary graph products of groups. In particular we establish the following Corollary to the construction used in the proof of the Main Theorem.

COROLLARY. Any graph product of finite groups is a Bridson group.

The term 'Bridson group' will be defined in the next section. The construction used to establish both results is an extension of a construction of M. Davis [6]. The curvature bounds follow from an application of the techniques of Moussang [15]. In establishing the Main Theorem we are essentially applying a specific technique for 'hyperbolizing' complexes analogous to Gromov's perturbation technique (see Sections 3.4 and 4.2 of [1 2]); for a detailed discussion of hyperbolizing complexes, see [4]. The spaces constructed here are known to be Tits buildings, by work in [7], where the Corollary to the Main Theorem is independently established.

2. Curvature and Siebenmann Complexes

We will be working with the geometry of piecewise Euclidean or hyperbolic complexes. We briefly review the necessary material; a good general reference for the geometry of such 'CAT(x ) ' spaces is [3].

DEFINITIONS. A geodesic in a metric space 34 is an isometry of an interval into 34, I ¢ ~ M. Let 34 be a metric space where, for each pair of points x, y E .M, there is a unique geodesic [0, d(x, y)] ¢ ,34 with ¢(0) = x and ¢(d(x, y)) = y. We denote the geodesic between x and y by Ix, y]. The space .M is CAT(-1) if given any triple of points {x, y, z} E .M and any point p E [y, z], and given any three points {x ~, y', z ~ } in the hyperbolic plane with sides [x ~ , y'], [y~, z ~] and [z ~ , x ~] of the same length as the corresponding sides of the triangle with vertices {x, y, z} in 34, then dM(x,p) < dT-t(x',p ~) where p' is the point on [y~, z ~] a distance d(y, p) away from y~. Similarly, 34 is CAT(0) if the same inequality holds when the triangle on {x, y, z} is compared to a triangle in the Euclidean plane.

It is not hard to show that any finitely generated group F which admits a discrete, cocompact action on a 1-connected CAT(-1) space is a word hyperbolic group. Recently, groups which admit discrete, cocompact actions on 1-connected CAT(0) spaces have been affectionately referred to as Bridson groups; viewing this as preferable to 'CAT(0) groups', we adopt this terminology.

Let S be a finite set of shapes; that is, S is a finite collection of compact spherical, Euclidean or hyperbolic polytopes. If X" is a cellular complex, then a geometric structure on ,g is an identification of the cells of X' with the shapes in some finite set of shapes S with the assumption that all the attaching maps are isometrics.

32 ~HN MEIER

For each finite set of shapes there is a sufficiently small e > 0, such that, for every vertex v, of any shape P, if the e-bail about v intersects a face F of P, then v is contained in the closure of F. The combinatorial link of a vertex v in ageometric complex X is thee-bal lB~(e) = {p E X [ d(p,v) = e}; if X is a simplicial complex this definition of link is equivalent to the standard definition. This complex can be given a geometric structure by (possibly) rescaling the inherited piecewise spherical metric so that each cell is isometric to a cell on the unit sphere. We will always assume the links are given this piecewise spherical metric.

It is a fundamental result that the large-scale curvature of a geometric complex is determined by the curvature of its shapes and the links of the vertices. We briefly sketch the important aspects of this result as it relates to the cubical complexes studied in this paper.

DEFINITION. The girth ofa piecewise spherical complex is the minimal length of a geodesic loop in the complex. (For finite complexes, the infimum of the lengths of geodesic loops is always acheived by an actual geodesic loop in the complex.)

THE LARGE LINK THEOREM 2.1 (Ballmann, Bridson, Gromov). Let X be a 1-connected, piecewise hyperbolic complex. If the links of the vertices of X all have girth greater than or equal to 27r, then X is CAT(- 1). If X is piecewise Euclidean with links of girth greater than or equal to 27r then X is CAT(0).

The link criteria in the Large Link Theorem seems to be fairly difficult to apply in dimensions above two. Thankfully, for cubical complexes, there is an equivalent combinatorial condition on the links of the complexes (see Sect. 4.2.C on pp. 122-124 of [12]).

DEFINITIONS. In a simplicial graph, a clique is a subgraph which is a complete graph. Let X be a simplicial complex. Then X is aflag complex if given any clique Q contained in X0), the 1-skeleton of X, there is a simplex S of X with Q = S(1). A simplicial complex satisfies Siebenmann's no-El-condition if each circuit of four edges bounds the union of two simplices along a common edge. Finally, a flag complex satisfying the no-El-condition is a Siebenmann complex.

Notice that the link of a simplex in a flag complex is itself a flag complex, and the link of a simplex in a Siebenmann complex is itself a Siebenmann complex.

Let X be a cubical complex. If each n-cell of X is given the metric structure of a unit Euclidean n-cube, then the links of the vertices will all inherit piecewise spherical structures with each simplex isometric to an 'all-right' simplex; that is, the simplex formed by intersecting the cone framed by the positive coordinate axes with the unit n-sphere in Euclidean (n + 1)-space.

WHEN IS THE GRAPH PRODUCT OF HYPERBOLIC GROUPS HYPERBOLIC? 33

THEOREM 2.2 (Gromov). Let £ be a flag complex (Siebenmann complex) with an all-right spherical metric structure. Then the girth of £ is greater than or equal to 27r (greater than 270.

3. The Direct Sum Simplex

To establish curvature bounds for our cubical complexes we need to examine the links of vertices in these complexes. In particular we will need to show that certain links in our complexes, those arising from direct sums of groups, are flag complexes.

Let A n be a simplex of dimension n, with vertices v~ labelled by the numbers A/" = { 1 , . . . , n + 1 }. If 27 C .N" then let F1- be the minimal face of A n containing {vi ] i 6 27}. We will assign groups to the faces of A n and then define a simplex of groups structure G(A n) based on A n. (For information on 'simplices of groups' or the more general 'complexes of groups', see [13].)

Assign the trivial group to the entire simplex and nontrivial groups GH-{i} to the codimension one faces FH-{i} of A n. For any other face Fz of A n, let the group associated to Fz be Gz = ®iczGH-{i}. This defines a simplex of groups structure on A n where the maps Gz --+ Gz,, for I ' C I, are the natural inclusion maps. Such a simplex of groups we refer to as a direct sum simplex of groups. One can think of this simplex of groups as a diagram of groups and monomorphisms where if Fz is a face contained in the boundary of Fz,, then Gz, injects as a proper subgroup of GI . For example, if the underlying simplex is a tetrahedron, and all the face groups are copies of Z2, then the edge groups will be Z2 x Z2 and the vertex groups will be (Z2) 3 ~ Z2 x Z 2 X g 2.

Simplices of groups arise most naturally from group actions on 1-connected complexes. Let ,1:' be a 1-connected simplicial complex and let G act on ,Y with X / G ~_ A n. Let G(A n) be the diagram of groups defined by the isotropy groups (and their inclusions) of the faces of a fixed copy of A s in 2(. Because ,%' is 1- connected, the original group G is the direct limit of this diagram of groups, or in Haefliger's terminology, the fundamental group of the simplex of groups, denoted g l [ G ( A n ) ] .

A simplex of groups is developable if it arises as the diagram of groups and monomorphisms induced by looking at the isotopy groups of a fundamental domain for some group action on a 1-connected complex (as above). By Sections 3.4 and 4.1 in [13], a simplex of groups is developable if and only if there is an injection of the diagram of groups into a target group F. If we let F : ®ieAcG.v-{i}, then clearly our direct sum simplex of groups embeds in F and hence it is developable. Further, by examining the presentation for the direct limit, it is easy to check that r = 7h [G(An)]. Thus we have the following result.

LEMMA 3.1 Let G( A ~ ) be a direct sum simplex of grou_ps as outlined above. Then G(A n) is developable and the fundamental group of G( A n) is F = OirJq 'GN_{ i } .

34 JOHN MEIER

Because our simplex of groups is developable, there is an action of F on some simplicial complex X with quotient X / F ~_ A n such that the diagram of groups G(A n) is isomorphic to the diagram of groups G~(A n) described by taking the isotropy groups of the faces of any simplex A n C X. In the example mentioned before, where the faces of a tetrahedron have groups of order 2 associated to them, X would be the decomposition of the unit 3-sphere in ~4, by 16 all right tetrahedra and 7rl [G(A3)] _~ (Z2) 4 is the group generated by the reflections in the four coordinate hyperplanes.

Let X be the 1-connected complex associated to a direct sum simplex of groups G(A n) and let 7) be a fundamental domain for the action of F = 7rl [G(An)]. Thus X -- F-7) and this fact gives us an easy description ofX. For each j EAf, the vertex vj of 7) is fixed by the group G{j). We denote this vertex of 7) by an (n + 1)-tuple of n minus signs and a single plus sign, ( - , . . . , + , . . . , - ) , where the plus sign occurs in the j th position. If v is any vertex in X, then v = g • ( - , . . . , + , . . . , - ) for some vertex ( - , . . . , + , . . . , - ) E 7) and some g E F. Further, because the isotropy group of vj is G{j) = ®iCj Gx-{ i ) , the group element g may be assumed to be

g = ( 1 , . . . , g j , . . . , 1) E GH_(1 ) × . . . × G~-_{j} × . . - × GH_{n+I ).

Thus, there is a one-to-one correspondence between vertices of X and pairs of (n + 1)-tuples of the form:

( 1 , . . . , g j , . . . , 1) • ( - , . . . , + , . . . , - ) ,

where the ' + ' occurs in the j th position and gj E GAf_{j). Similarly, the edges of A' correspond to pairs of (n + 1)-tuples

( 1 , . . . , g i , . . . , g j , . . . , 1 ) . ( - , . . . , + , . . . , + , . . . , - )

and in general an m-simplex of X will correspond to a pair of (n + 1)-tuples: the second (n + 1)-tuple contains m plus signs and (u + 1 - m) minus signs, and the first (n + 1)-tuple contains m entries from codimension one face groups in each position marked by a plus and the remaining positions are filled by 1 's. We define the type of a simplex to be the associated (n + 1)-tuple ' ( + , . . . , + ) ' of plus and minus signs.

Let the following pair of (n + 1)-tuples

( h l , . . . , h n + l ) " ( e l , . . . , e n + l )

represent an m-simplex A in X. If ej = + for some j , then the (m - 1)-simplex

. . . h' . . . ( h i , , n+ l ) " (E:i, , gn+l )

t ~ - is a codimension defined by h~ = hl and e i = ei for i ~ j and hj = 1,ej = 1 face of the original m-simplex. In general, if 37 C {i E A f [ el = + ) , then the simplex

(h~ ' , . . . , " . . . hn+l)" " ~n+l)~

WHEN IS THE GRAPH PRODUCT OF HYPERBOLIC GROUPS HYPERBOLIC? 35

where h~' = hi and e~' = el for all i ~ 37, and h~' = 1 and e~' = - for each i E 37, is a codimension 1371 face of the original simplex A.

PROPOSITION 3.2 The simplicial complex P( described above is a flag complex. Proof If Q is a clique in the 1-skeleton of A', then the vertices of Q are all of

distinct types. If we let

Q(o) = { ( 1 , . . . , g i , . . . , 1 ) . ( _ , . . . , + , . . . , _ ) i i e Z}

for some Z CAf, then it follows from the description above Q is the 1-skeleton of the simplex

(hl,...,hn+l). ( c 1 , . . . , c n + 1 ) ,

where hi = gi i f / E Z, hi = 1 otherwise, and similarlyei = + i f i E Z andei = - otherwise. []

4. The Cubical Complex

The proof of the implication ~ in the Main Theorem is contained in the comments in the introduction. To show ~ , we construct a cubical complex admitting a discrete cocompact action by the graph product Gf~ . In the next section we establish the hyperbolicity of this complex. In order to make the steps of the construction more understandable, we recommend that the reader periodically consult the example given in the concluding section.

Our construction is valid for any graph and any assignment of groups to the vertices of this graph. In general this process will create a CAT(0) cubical complex on which the graph product G9 acts. However, this action will not always be discrete, and this space is not generally Gromov hyperbolic. Since our primary goal is to create a CAT(-1) complex on which the graph product of groups GSt-9 acts, we have written most of this and the next section assuming 9r9 satisfies the hypotheses of the Main Theorem.

DEFINITIONS. Every simplex has a natural cubulation. The cubulation of a 1- simplex is its barycentric subdivision and a cubulated n-simplex is isomorphic to the cone of the cubulation of its (n - 1)-skeleton. The cubulation of a simplicial complex is the cubical complex created by cubulating each simplex. Notice that the cone of a cubulated n-simplex has the combinatorial structure of a subdivided (n + 1)-cube (Figure 1).

In a cubical complex, the closed star of a cell c is the closure of the union of cells which contain c.

A cell in a complex is maximal if it is not properly contained in the closure of any other cell of the complex. Note: a maximal cell is not necessarily a top dimensional cell.

o$-. (b) .." : ........

36 JOHN MEIER

Figure 1. (a) Cubulated simplex; (b) cone over a cubulated simplex.

Form a flag complex whose 1-skeleton is isomorphic to 3c¢ by filling in each clique in ~0 by a simplex. Denote the resulting simplicial complex by ~ * . Because of condition (iii) in the statement of the Main Theorem, Fa* is a Siebenmann complex.

Let 79 be the cone of the cubulation of )re , with cone point e. The finite cubical complex 79 will be the fundamental domain for the action of GSc¢ on a CAT(-1) cubical complex C:r-. Notice that the barycentric subdivision of.)ra embeds into the l-skeleton of 79 and we refer to the vertices of ~ , thought of as vertices of 79, as the primary vertices of 79. If v is the barycenter of a maximal simplex in )c~, then v is a clique vertex of 79 and any vertex in 79 which is not the cone point or a clique vertex is a join vertex.

If v is a primary vertex of 79, let My be the closed star of v in the cubulation of .T'¢*, thought of as a subcomplex of 7). In the case where G~¢ is a right-angled Coxeter group, one thinks of the Mv's as 'mirrors' contained in 79 across which 79 is 'reflected' to create a 1-connected cubical complex [6]. In the present situation a better metaphor is that the Mv's are 'hinges' about which are attached [Gvl copies of / ) .

To each point p in 79 associate the subgraph G; of ~r¢, which is generated by the vertices {v C 3r¢ [ p C M~}. Thus for each point there is an associated group G~p. For example, if p is a clique vertex, that is p is the barycenter of a maximal simplex A, then A(1) is a (subdivided) clique Q C ~G. The group G~; is then the graph product of groups based on the subgraph Q; hence G~p is the direct sum of groups G~ where v E Q. For every point p the subgraph Gp is a complete graph, and the vertex groups in ~G are finite, so GGp is a direct sum of finite groups. As a convention, if p is contained in no My, then its associated graph is the empty graph and its associated group is { 1}.

Let C~- be the cubical complex formed by taking the quotient of G ~ × 79 by the relations (g, x) ,,o (h, y) if and only if x = y and g- lh E GGx. (Here we give

WHEN IS THE GRAPH PRODUCT OF HYPERBOLIC GROUPS HYPERBOLIC? 37

GSt-¢ the discrete topology and C~- the product topology.) This complex admits a natural action by G.Ta via 9" (h, z) = (gh, z). The quotient of C7 under the action of G.T9 is compact and isomorphic to 79 and the stabilizer of a cell tr is the group GGp for any point p contained in the interior of tr. Since G~p is finite and 79 is compact, the action of GSr9 on C:r- is discrete and cocompact. That C~- is simply connected essentially follows from Corollary 10.2 of [6].

5. The Curvature of C~-

We give C~- a CAT(-1) metric structure in order to establish that G.T~ is a word hyperbolic group. In order to do this we first need to examine the combinatorics of the links of the vertices of C5.

If v is any vertex in C~- then v is in the orbit of the cone point c of 79, a clique vertex of D or some join vertex of D. If v happens to be in the orbit of the cone point c then the link of v is simply the Siebenmann complex .TO*. If v corresponds a clique vertex then its link is a direct sum simplex defined by setting the codimension 1 face groups of this direct sum simplex equal to the vertex groups Gv where v is a vertex in .T~ associated the the clique forming the 1-skeleton of which v is the barycenter. It follows by Proposition 3.2 that the link of v is a flag complex.

If v is a join vertex, then let v --+ c denote the simplex in U~ ( = the link of the cone point c) corresponding to the cube in D with antipodal vertices v and c. The join vertex v is also a vertex in at least one maximal cube in D. Let q be the clique vertex corresponding to one such maximal cube and let v --+ q be the simplex in the link of q corresponding to the cube with antipodal vertices v and q. The link of v is the join of the link o fv ~ c in .Ta with the link of v --+ q in the link ofq. (This is independent of choice of clique vertex q.) It follows that the link of any vertex in the orbit of a join vertex is the join of a flag complex and a Siebenmann complex.

To construct a hyperbolic structure on C7 we borrow the hyperbolic structure that Moussang uses for his cubical complex associated to the right-angled Coxeter group defined by the graph f g . Because Moussang's thesis is not yet published, we briefly run through the details. Let O be the origin in hyperbolic n-space, where n is the maximal number of vertices in a clique of f g , and let To be the tangent space at the origin. Fix n mutually orthogonal vectors U = {ui [ 1 < i < n} in To. For any subset W C U, let Cw be the image under the exponential map To --+ H ~ of the cone in To framed by W. For each e > 0 there exists a point p~ E H ~ which is a distance e away from each codimension one face of Cu. For each subset W C U let Pw be the point in Cw which is closest to p~. By definition, pc = O and Pu = P~. Let B~ be the cube formed by taking the convex hull of {Pw [ W C U}. Notice that the link of O in B~ inherits an all-right metric structure and that as

--+ 0, the link of p~ in B~ comes arbitrarily close to an all-right simplex. Each maximal cube in 7) contains the cone point c of D and a unique clique

vertex. Give each maximal cube in 79 the metric structure of a cube B~, of the appropriate dimension, where the point O in B~ is identified with the clique vertex

3 8 JOHN MEIER

and the point Pe is associated to the cone point c. Because the link of e is a Siebenmann complex, by Theorem 2.2 there is a sufficiently small e > 0 such that the link of e with the piecewise spherical metric induced by B, has girth greater than or equal to 2rr. Extend this metric structure on 79 to the complex C7 by the action of G.T9. Because the link of any vertex in the orbit of a clique vertex is a flag complex, and Gromov's Theorem 2.2 insures these have girth 27r, the link criteria of the Large Link Theorem is satisfied at these vertices.

The only remaining vertices to check are the join vertices. However the links of these vertices are joins of a link in f~* with a link in a flag complex. It follows by Corollary 5.6 and the proof of Theorem 17.1 in [15] that these links also have girth 27r.

PROPOSITION 5.1 Let ~ be any graph where no full subgraph is a circuit of order four. If all the vertex groups associated to G are fnite, GG is a word hyperbolic group.

As was mentioned earlier, the fact that the graph ~ has no full subgraph equal to a cycle of order 4 is not necessary for the construction of the cubical complex C~-. In particular, because the cliques of f 9 are filled in forming 5t'9 *, the link of every vertex in C7 is a flag complex reguardless of the structure of the original graph ~ . Thus, if each n-cell in Cy is given the metric structure of a unit Euclidean n-cube, then C7 becomes a CAT(0) space by Theorem 2.2 and the Large Link Theorem. If each group associated to a vertex in 5c9 is finite, then the action of GSc~ on C:r- is discrete and cocompact; hence we have established the Corollary to the Main Theorem.

COROLLARY 5.2 Any graph product of finite groups is a Bridson group.

6. Finishing the Proof

To finish the proof of the Main Theorem, let v be a vertex of ~ - Fa, hence the associated group Gv is an infinite word hyperbolic group. By conditions (i) and (ii), Z;~ C G is a complete graph which is contained in the subgraph ~¢; hence G£,~ is a direct sum of finitely many finite groups. By the decomposition established by E. Green, the graph product based on the subgraph of G generated by )r e U {v} decomposes as the free product of G~ x G£~ and G.T¢ amalgamating G£v. However, by hypothesis, Gv is word hyperbolic, and because GL;~ is finite, G~ x G/2v is also word hyperbolic. By Proposition 5.1 we know that GSC¢ is word hyperbolic. Thus the graph product G{~¢ U {v}} is the amalgamation of two word hyperbolic groups amalgamating a finite group, hence G{~'G U {v}} is word hyperbolic.

WHEN .IS THE GRAPH PRODUCT OF HYPERBOLIC GROUPS HYPERBOLIC?

D

39

C ~ ' DF ,

B~-- - O F

A

Figure 2. The graph .T'g

Figure 3. The complex D.

Repeating this proceedure for each remaining vertex v 6 g - ~a amounts to taking free products of word hyperbolic groups amalgamating finite subgroups, hence the graph product GG is a word hyperbolic group. []

40 JOHN MEIER

Figure 4. Part of the cubical complex C:~, I.

Figure 5. Part of the cubical complex C:r-, II.

7. Example

Let the graph G be a circuit composed of 5 edges with two additional edges added to form a triangle. All of our vertex groups will be finite, so in this case G = ~'9 (Figure 2).

Let all of the vertex groups be copies of Z2, except for the top vertex D which will have Z3 associated to it. For this example the cubical complex D is the union of a cubulated pentagon with a 3-cube joined along one of the 2-cubes forming the pentagon (Figure 3). The points indicated are the primary vertices of D; the primary vertex corresponding to A is not visible since it is on the bottom of the 3-cube.

The darkened portion ofT? is the hinge 3rib corresponding to the primary vertex B.

In forming the cubical complex C, the 3-cube will be reflected in its three external faces to form a larger cube; similarly D will be rotated about the hinge AdD to form three copies of D joined along AdD (Figures 4 and 5).

WHEN IS THE GRAPH PRODUCT OF HYPERBOLIC GROUPS HYPERBOLIC? 41

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