Cat(0) and Cat (-1) dimensions of torsion free

CAT(0) AND CAT(-1) DIMENSIONS OF TORSION FREE HYPERBOLIC GROUPS NOEL BRADY1AND JOHN CRISP2 Abstract. We show that a particular free-by-cyclic group has CAT(0) dimension equal to 2, but CAT(-1) dimension equal to 3. We also classify the minimal proper 2-dimensional CAT(0) actions of this group; they correspond, up to scaling, to a 1-parameter family of locally CAT(0) piecewise Euclidean metrics on a fixed presentation complex for the group. This information is used to produce an infinite family of 2-dimen- sional hyperbolic groups, which do not act properly by isometries on any proper CAT(0) metric space of dimension 2. This family includes a free-by-cyclic group with free kernel of rank 6. Contents 1. Introduction 2 2. Definitions and background 4 3. The group G=ha, b|aba2=b2i: geometric structures 5 3.1. The free-by-cyclic structure 5 3.2. The 3-dimensional CAT(-1) structure 7 3.3. The 1-parameter family of CAT(0) structures 8 4. Proof of Theorem 1 (2): an analogue Flat Torus Theorem 15 4.1. Finding a minimum for the combined displacement 17 4.2. Angle measurements 18 5. The hyperbolic versus CAT(0) problem: Theorem 2 21 References 24 Date: July 12, 2005. 1Partially supported under NSF grant no. DMS-0124344. N. Brady thanks the Centre de Recerca Matematica for support and hospitality. 2Supported by a CNRS postdoctoral research assistantship in 2001 and thanks the IMPA ENS Lyon for hospitality during that period. 1 2 N. BRADY AND J. CRISP 1. Introduction One of the foundational problems in the theory of hyperbolic groups is to determine the relationship between coarse and continuous notions of negative curvature. Specifically, one is interested in the relationship between coarse notions such as Gromov’s δ-hyperbolicity, and the more continuous notions due to Alexandrov and Toponogov of CAT(0) and CAT(-1) metric spaces. It is known that if a group acts properly and cocompactly by isometries on a CAT(-1) metric space (or on a CAT(0) metric space which contains no isometrically embedded flat planes [9]) then the group is hyperbolic in the sense of Gromov. It is still an open problem as to whether all hyperbolic groups act properly and cocompactly by isometries on CAT(0) metric spaces. The main results of this paper show that if one is trying to find a proper CAT(0) or CAT(-1) metric space on which a given torsion free hyperbolic group acts properly and cocompactly by isometries, then the dimension of the space may have to be be strictly greater than the geometric dimension (usual topological dimension) of the group. We find that this is so in the case of hyperbolic free-by-cyclic groups. The basic example upon which everything else is built is an F3o Z group which has CAT(0) dimension equal to 2, but has CAT(-1) dimension equal to 3. Furthermore, there is a one-parameter family of CAT(0) piecewise Euclidean 2-complexes associated to this group with the following property: every 2-dimensional proper CAT(0) space on which this group acts properly isometrically contains an isometrically embedded scaled copy of one of these 2-complexes. Theorem 1. The group Gwith presentation ha, b |aba2=b2iis of the form F3o Z, is hyperbolic, and admits a compact locally CAT(-1) 3-dimensional K(G, 1). Furthermore, there is a one-parameter family {Kt}tof compact, locally CAT(0), piecewise Euclidean 2-complexes with the following properties: (1) Each Ktis a K(G, 1). In particular, Ghas CAT(0) dimension equal to 2. (2) Every proper CAT(0) space of dimension 2 on which Gacts properly discontinuously by isometries contains an isometrically embedded, G-equivariant copy of the universal cover of some Kt(up to a constant scaling of the metric on Kt). In particular, Gdoes not act properly isometrically on a proper CAT(-1) space of dimension 2, and so Ghas CAT(-1) dimension equal to 3. Property (2) in Theorem 1 can be viewed as a first (weak) hyperbolic analogue to the Flat Torus Theorem. Here is the analogy. Say that a group CAT(0) AND CAT(-1) DIMENSIONS OF HYPERBOLIC GROUPS 3 Gacts minimally on a CAT(0) space Xif Xcontains no proper convex G- invariant subspace. Then the Flat Torus Theorem states that any minimal proper semi-simple action of Znon a CAT(0) space is an action by translations on En. Our Theorem 1 (2) states that the minimal 2-dimensional CAT(0) actions of the group Gare classified by the model complexes Ktfor t∈R. The analogy is “weak” in the sense that we impose the dimension restriction. On the other hand, we do not suppose the actions are semi-simple. This mimicks the statement of the Torus Theorem of Fijuwara, Shioya and Yamagata [13] which includes the dimension restriction, but does not require semi-simplicity. In fact we shall use Proposition 4.4 of [13] in order to find a minimum for combined displacement function without recourse to any cocompactness hypothesis. The Flat Torus Theorem has been very useful in proving that certain groups are not CAT(0) [14]. The groups typically contain a Z2subgroup, together with a lot of conjugation relations, so that any putative, nonpositively curved K(π, 1) for them will contain an impossibly shaped flat torus. In the present case the Ktcomplexes play the role of flat 2-tori. They can be used to construct groups which will not be CAT(0) in dimension 2, because extra conjugation relations will somehow contradict the geometry of the Kt. For example, we have the following theorem. Theorem 2. There is an infinite family of hyperbolic groups of geometric dimension 2, which do not act properly by isometries on any proper CAT(0) metric space of dimension 2. This family includes infinitely many free-by-cyclic groups, one of which is F6o Z. Remark 3. This theorem offers the first conclusive proof that in tackling the question of whether all (torsion free) hyperbolic groups are CAT(0), one is obliged to look for CAT(0) structures above the geometric dimension of the group. This is the case even within the class of hyperbolic free-by-cyclic groups. It was already known that it is easier to find CAT(0) structures for hyperbolic groups if one looks above the geometric dimension. The work of Wise [20] and of Brady-McCammond [7] all find high dimensional CAT(0) piecewise Euclidean cubical structures for various classes of hyperbolic groups (certain small-cancellation groups, and certain families of ample twisted face pairing 3-manifold groups). The results of this paper imply that it is not only easier, but that in some cases it is also necessary to look above the geometric dimension in the search for CAT(0) structures. By the work of Bridson [10] and of Brady-Crisp [5] (see also [12, 13]) one knows that the minimal dimension of a CAT(0) structure for a CAT(0) group, may be strictly greater than its geometric dimension. However, all 4 N. BRADY AND J. CRISP these papers used some version of the flat torus theorem, and made heavy use of periodic flats in 2-dimensional CAT(0) spaces. The key idea in the current paper is to find a very special hyperbolic group, and corresponding 2-complexes, which play the role of Z2subgroups and flat 2-tori. We wish to thank Luisa Paoluzzi, Robert Roussarie, Sylvain Crovisier and Christian Bonatti for (collectively) bringing to our attention the full variety of 2-dimensional structures Kt, and for several helpful observations and simplifications of the arguments of Section 3.3. We also wish to thank Lee Mosher and Leonid Potyagailo for raising the questions mentioned in Section 3.1. 2. Definitions and background Ametric space Xis said to be proper if every closed ball Br(a) in X is compact. An action of a group Gby isometries on a metric space Xis said to be proper if for each x∈Xthere is an open ball Br(x) about x (r > 0) such that g(Br(x))∩Br(x) is nonempty for only finitely many g∈G. This is slightly more restrictive than the usual notion of a proper action on a topological space, namely that the set {g∈G:g(K)∩Knonempty } is finite for any compact K⊂X, but the two notions are equivalent for actions on proper metric spaces – see Remarks on page 132 of [11]. We refer to [11] for details on CAT(κ) spaces, for κ≤0; metric spaces of global non-positive curvature bounded above by κ∈R. Let gbe an isometry of a CAT(κ) space X,κ≤0. The translation length of gis defined as l(g) = inf{d(x, gx) : x∈X}. The isometry gis said to be semi-simple if it attains its translation length at some point of x. In this paper we make no assumptions on the semi-simplicity or otherwise of our group actions. This is in contrast to previous works [5, 10, 12] where semi-simplicity is assumed (because it is needed to apply the usual Flat Torus Theorem). In these cases this hypothesis can be removed by using instead the 2-dimensional Torus Theorem of [13], at the expense of supposing that action is on a proper CAT(0) space. In this paper we shall use the notion of “geometric dimension” introduced by Bruce Kleiner in [18] for the class of metric spaces with curvature bounded above in the sense of Alexandrov [1], termed CBA spaces. These spaces include all complete CAT(κ) spaces (κ∈R). Associated to any point pin a CBA space Xis the space of directions ΣpX, which is known to be a complete CAT(1) space (see [19]). The geometric dimension of a CBA space is “the largest number of times we can pass to spaces of directions without getting the empty set”– more precisely, the smallest function GD :{CBA spaces} → N∪{∞} such that GD(X) = 0 if Xis discrete, and otherwise GD(X)≥1 + GD(ΣpX) for all pin X. CAT(0) AND CAT(-1) DIMENSIONS OF HYPERBOLIC GROUPS 5 In [18], Kleiner shows that the geometric dimension is a lower bound for the usual covering dimension (defined in general for topological spaces, see [17]). Moreover, he remarks (on p.412) that these two dimensions coincide for separable CBA spaces, which include proper CAT(κ) spaces. Since it will be useful later, we recall that the space of directions ΣpX is defined as the space of all equivalence classes of geodesics emanating from p, where two geodesics are said to be equivalent if the Alexandrov angle between them is zero, and carries a metric induced by the Alexandrov angle. 3. The group G=ha, b|aba2=b2i: geometric structures In this section we prove all of the statements contained in Theorem 1 with the exception of part (2), which we defer until the next section. The work is broken into three subsections: in (3.1) we show that the group Gis F3o Z, in (3.2) we exhibit a 3-dimensional CAT(-1) structure, and in (3.3) we introduce the one-parameter family of 2-dimensional CAT(0) structures. We shall give a Morse theory argument that the group is free-by-cyclic. This will easily extend to show that certain HNN-extensions with base Gand Z edge groups are also free-by-cyclic (see Proposition 19 of Section 5). 3.1. The free-by-cyclic structure. The group Ghas presentation ha, b |abaa =bb i, and the corresponding presentation 2-complex has one vertex (labeled v), two 1-cells (labeled aand b), and a single hexagonal 2-cell (labeled by the relation). Any map of Gto Ztakes the generators aand bto integers Aand B respectively which satisfy the equation 3A+B= 2Bor B= 3A. Thus we may assume that ais taken to a generator of Zand bto three times this generator. We can realize this homomorphism topologically by a map from the presentation 2-complex to the circle (with one 0-cell and one 1- cell). This map sends the vertex vto the base vertex of S1and maps a once around the circle, and bthree times around the circle. Extend this map linearly over the 2-cell. This lifts to a Morse function on the universal cover. Figure 1 shows how a typical 2-cell of the universal cover looks with respect to the Morse function. The preimage of the base vertex of S1is a graph in the 2-complex, and is shown as the graph Γ in Figure 1. The vertices [b/3] and [2b/3] denote points which are respectively 1/3 and 2/3 along the edge b, and which map to the vertex of S1. Note that π1(Γ) is F3. One can check that the preimage of a generic point of S1will be a graph, ∆, with 4 vertices, and 6 edges. As the generic point on the circle moves 6 N. BRADY AND J. CRISP towards the base vertex, an edge of the preimage graph collapses to a point, 1giving a homotopy equivalence with the graph Γ. ya b b a b a v G x PSfrag replacements Γ a b [b/3] [2b/3] v Figure 1. The Morse function on the 2–cell of the F3o Z group, and the level set Γ through the vertex v. Thus the presentation 2-complex of Gcan be viewed as a graph of spaces whose underlying graph is the circle (with one vertex and one 1-cell), whose edge space is ∆, whose vertex space is Γ, and whose maps are the homotopy equivalences ∆ →Γ obtained by collapsing particular single edges of ∆. Thus Gis isomorphic to the fundamental group of this graph of spaces, and so is F3o Z where the monodromy automorphism is obtained by composing the “ascending” homotopy equivalence ∆ →Γ with the inverse of the “descending” one. It is a good exercise to work out this automorphism explicitly from the graph of spaces description; it is a “change of tree” automorphism, although it is not a change of maximal trees. Here we give an explicit description of the automorphism in terms of the original presentation of G. The group Gis an extension of the free group F3hx, y, ziby Zwhere Zacts via the automorphism ϕ:x7→ y7→ z7→ x−1y; that is Gis the HNN-extension F3∗ϕ. The automorphism is just conjugation by a. Putting x=a−2bx−1(so b=a2xa) the relation aba2=b2is rewritten a3xa3=a2xa3xa , 1This collapsing edge corresponds to either the ascending or the descending link of the Morse function. See Bestvina-Brady [3] for terminology, or Brady-Miller [8] where the connection between Morse theory and free-by-free groups is made explicit. CAT(0) AND CAT(-1) DIMENSIONS OF HYPERBOLIC GROUPS 7 which easily rearranges to ϕ(x) = xϕ3(x), or rather ϕ3(x) = x−1ϕ(x), where ϕdenotes conjugation by a. Thus Gis isomorphic to the given HNN-extension. The automorphism ϕis exponential, but with a very low expansion rate: λ= 2.325 = (the solution to λ3=λ+ 1). This may give a heuristic explanation for the reluctance of Gto act on a CAT(−1) 2-complex. We do not know whether the extensions F3∗ϕnfor n > 1 (which are subgroups of index nin G) are CAT(−1) in dimension two. This latter problem was raised by Lee Mosher and is closely related to the following question suggested by Leonid Potyagailo. Question 1. Does every 2-dimensional word hyperbolic group contain a finite index subgroup which acts properly cocompactly by isometries on a 2-dimensional CAT(−1) proper metric space? 3.2. The 3-dimensional CAT(-1) structure. Let Pdenote a regular solid octahedron of “small” volume in hyperbolic 3-space, as illustrated in Figure 3.2. We label the vertices 1,2, .., 6 as indicated in the figure, and define the piecewise hyperbolic 3-complex Mto be obtained from Pby identifying the pair of faces labelled (1,4,6) and (6,3,5) to a single face A, and the pair (1,5,2) and (2,4,3) to a single face B(respecting the order of vertices in each case). The remaining four faces are left open. Choose a basepoint in the interior of Pand define oriented paths aand bin Mpassing through the faces Aand B, respectively, as indicated in the figure. One easily checks that the loops aand bgenerate π1(M) subject to the single relation abaa =bb. That is π1(M)∼ =G. (In fact the K(G, 1) complex Kdiscussed the next subsection can be embedded in Mas a “2- spine”– the complex Kis a deformation retract of M, showing that Mis also a K(G, 1)). The complex Mhas a single vertex vwith link Lk(v, M) as illustrated in Figure 3.2. This is a Moebius band composed of six spherical quadrilaterals with sidelengths all equal to π/3−ε1and diagonals all of length π/2−ε2 where ε1, ε2both tend to zero as the chosen volume of the octahedron P tends towards 0. A systole for Lk(v, M) is shown in bold in the figure. If P were chosen Euclidean, then the length of this systole would be 4.(π/3)+2µ where µ(the length of the segment crossing quadrilateral 2) lies strictly between π/3 and π/4 (in fact µ > 72◦). In the small volume hyperbolic case the systole measures 4.(π/3) + 2µ−εwhere εalso tends to 0 with the volume of P. For sufficiently small choice of volume of Pthis value is larger than 2πand Mis a locally CAT(-1) space. This also gives a further way of seeing that Mis indeed a compact K(G, 1) for our group G. We refer the reader to [6] for further details concerning determination of the systole in Lk(v, M) and the calculation of its length. 8 N. BRADY AND J. CRISP 1 2 3 4 5 6 b b a a 1 2 3 4 5 6 b b a b a a −1 −1 Figure 2. The 3-dimensional piecewise hyperbolic complex M=P/ ∼, and its vertex link Lk(v, M). 3.3. The 1-parameter family of CAT(0) structures. Let Kdenote the presentation complex defined by the 1-relator presentation G=ha, b |abaa = bb i. The associated Cayley complex e K(the universal cover of K) has been previously studied by both Haglund [15] and Ballman and Brin [2]. It is one of the two completely regular simply connected polyhedra which can be built out of regular hexagonal cells in such a way that every vertex link is a complete graph on 4 vertices. The other is the Cayley complex associated to the one-relator presentation with relation baa =abb, which defines the Geisking 3-manifold group. A portion of the complex e Kis illustrated in Figure 3. Note that the band of hexagons immediately surrounding the central one in the figure is twisted, so that their union is a Moebius band rather than an annulus as in the case of the Geisking complex. CAT(0) AND CAT(-1) DIMENSIONS OF HYPERBOLIC GROUPS 9 The complex e Kvery naturally admits a CAT(0) metric in which each cell is a regular Euclidean hexagon. Consequently, the quotient presentation 2-complex Kis a locally CAT(0) K(G, 1), and the group Gtherefore has CAT(0) dimension equal to 2. Moreover, since this Haglund, Ballman- Brin complex e Kdoes not contain any isometrically embedded flat planes (because of the twist), we have another way of concluding that the group Gis hyperbolic. 2 a b a b Note: twist in outer band of hexagons a b PSfrag replacements b a Figure 3. The Cayley complex for ha, b |aba2=b2i. 2In [16], M. Kapovich considered the full isometry group Isom( e K) of e Kand showed, by using the existence of torsion elements, that Isom( e K) does not act properly isometrically on any CAT(−1) space. This line of argument was pursued in [6] for this and various similar examples. However the techniques used here to study the torsion free subgroup Gare necessarily quite different and we have not as yet succeeded in extending them in this way to other examples. 16 N. BRADY AND J. CRISP d(x, b(x)). This is a straightforward consequence of Lemma 10 below, since here Gis a torsion free hyperbolic group of cohomlogical dimension 2. The second step involves constructing a map e K→X. Let Γ denote the 1- skeleton of e K(the Cayley graph of Gwith respect to {a, b}) and let vdenote a base vertex in Γ. Given any point x∈Xwe may define a continuous map ϕx: Γ →Xby sending vto x, extending G-equivariantly on the vertex set of Γ and then mapping each edge to the (unique) geodesic joining the images of its endpoints. This construction also leads to a natural choice of “lengths” for each edge in Lk(v, e K). For simplicity we write L= Lk(v, e K) and Σ = ΣxX, the space of directions at xin X. We recall that Lis simply the complete graph on 4 vertices. If pdenotes a vertex of L, determined by the edge esay, then we write pfor the direction in Σ determined by the geodesic segment ϕx(e). We now assign to each edge (p, q) in La “length” given by the distance between pand qin Σ. Note that, since it is possible that p=q, which endows the edge (p, q) with zero length, this choice determines a pseudo-metric, rather than a metric, on L. We shall write Lx to denote the graph Lequipped with this pseudo-metric. We shall prove further on (see Lemma 12 and the Remark which follows it) that if the point xis chosen so as to minimise the combined displacement function fthen Lxturns out to satisfy the link condition for a CAT(1) metric graph: each circuit has length at least 2π. On the other hand, the six edge lengths in Lxappear as the angles of a “geodesic hexagon” Cin X(take the image of any hexagonal circuit in Γ which bounds a 2-cell of e K). Nonpositive curvature in Ximplies that the sum of these angles is at most 4π(see[12], Lemma 1, for example). It now follows by Lemma 5 that the total of all six angles is exactly 4πand that each simple circuit in Lhas length exactly 2π. (The above arguments apply even when there are zero length edges in L). By the Flat Triangle Lemma [11], it now follows that the geodesic hexagon Cactually bounds a genuine convex (but possibly degenerate) 2-dimensional Euclidean hexagon Hisometrically embedded in X. By properness of the action of Gon Xthis hexagon cannot degenerate on to an interval (for then the orbit of xwould lie on a single line!), so has non-empty interior and nonzero angles. We now choose a G-equivariant metric on e Kby letting each 2-cell be isometric to the hexagon H, and extend the map ϕxto a map ϕx:e K→Xwhich is locally isometric on the interior of 2-cells. Note that the link of each vertex in e Kis isometric to Lxand CAT(1). Thus, by Proposition 4, e Kequipped with this metric is (up to scaling) G-equivariantly isometric to the universal cover of one of the model complexes Ktfor some t∈R. In particular, all edge-lengths in the link Lare strictly less than π. CAT(0) AND CAT(-1) DIMENSIONS OF HYPERBOLIC GROUPS 17 It now only remains to show that ϕxis locally isometric at each vertex. For this we need to prove that the induced map ϕ:L→Σ is π-distance preserving: if dΣ(p, q)< π then dL(p, q) = dΣ(p, q). Since Lhas diameter equal to πit suffices to show that ϕ:L→Σ is an isometric embedding. Let p1, p2, p3be three vertices of L, spanning a triangle ∆ with edge lengths φ1, φ2and φ3. We have that each length φilies in the range (0, π), and φ1+φ2+φ3= 2π. By Lemma 12, the image ϕ(∆) (in Σ) of the triangle ∆ is a closed geodesic of length 2π. Thus ϕmaps the triangle ∆ isometrically into Σ (it is a locally isometric embedding of a space of diameter πinto a uniquely π-geodesic space). Since any two points in L lie in a common triangle ∆ it now follows that ϕ:L→Σ is an isometric embedding, as required. ¤ 4.1. Finding a minimum for the combined displacement. In this section we prove the following result: Lemma 10. Let Xbe a proper CAT(0) space, g1, g2, . . . , gna finite collection of isometries of Xwhich generate a group Gacting properly on X, and λ1, λ2, . . . , λnstrictly positive real numbers. Then either (1) there exists a point in Xwhich minimises the “combined displacement function”: f:X→R+:f(x) = n X i=1 λid(x, gi(x)) or (2) the group Gfixes a point x∞on the ideal boundary ∂X of X. If Ghappens to be a word hyperbolic group with finite K(G, 1) then case (2) above implies that either G∼ =Zor cd(G)≤dim(X)−1. Proof. In the first instance, if fis a proper map then it clearly achieves a minimum. Supposing otherwise, and using the fact that Xis proper (and so X∪∂X is a compact space – see, for example, Exercise II.8.15(2) of [11]), one can find a sequence of points {xk}Nwhich converges to a point x∞in ∂X yet such that the sequence {f(xk)}is bounded. It follows that each gi fixes x∞and so (2) holds. Now suppose that Gis a hyperbolic group with finite K(G, 1) (ie: G is torsion free). In particular, Gis finitely generated. By considering the action of Gon the horofunctions at x∞we deduce an exact sequence H→G→Z where His the subgroup of elements which act by leaving invariant every horosphere at x∞. 18 N. BRADY AND J. CRISP Suppose firstly that the map G→Zis nontrivial. By properness of the action of G, any two elements which map nontrivially to Zmust share a common power. Thus all elements of G−Hleave fixed a common pair of points {p, q}in ∂G. Moreover Hmust also fix the pair {p, q}(since it conjugates elements of G−Hto elements of G−H). But this implies that Gis virtually Z, or rather Zsince it is torsion free. We may now suppose that G∼ =Hand acts by leaving invariant all horospheres at x∞. But then our conclusion that cd(G)≤dim(X)−1 follows directly from Proposition 4.4 of [13]. ¤ 4.2. Angle measurements. In this section we wish to use an idea from elementary calculus: that the “rate of change” of a function in any direction from a local minimum is never negative. The functions that we consider are linear combinations of distance functions in a CAT(0) space. For these reasons we introduce the following: Lemma 11. Let Xbe a CAT(0) space, and [p, q]a nontrivial geodesic segment (d(p, q)>0), let γ: [0, ²]→Xdenote a nontrivial constant speed geodesic with γ(0) = p, and let θdenote the Alexandrov angle at pbetween [p, q]and γ. Let f: [0, ²]→Rbe the (necessarily continuous) function such that f(t) = d(γ(t), q). Then lim t→0;t>0 f(t)−f(0) t=−cos θ . We refer to the above limit as the derivative of fin the direction of γ. Remark. The above Lemma asserts, if you like, the existence of a directional derivative in the first variable of the distance function, which is defined, for a point (p, q)∈X×X, over the space of directions ΣpXat pin X. Proof. Note firstly that the lemma is precisely true in the Euclidean plane E2. Now, given the general situation described above, choose in E2points bp,bqsuch that dE(bp, bq) = d(p, q), and a geodesic bγfrom bpsuch that the Alexandrov angle at bpbetween bγand [bp, bq] equals θ. This configuration determines the function b f(t) = dE(bγ(t),bq) where b f(0) = f(0) = f0say. By one version of the comparison axiom (Proposition II.1.7 (5) of [11]), we have that b f(t)≤f(t) for all t∈[0, ²]. Since b fis a convex function we then have f0−tcos θ≤b f(t)≤f(t) and hence f(t)−f0 t≥ −cos θfor all t∈[0, ²]. Fix s∈(0, ²] and let ∆0(p0, q0, r0) denote the Euclidean comparison triangle for the triangle in Xwith corners p, q and r=γ(s). That is dE(p0, q0) = d(p, q), etc... Let θsdenote the angle in E2between the sides of ∆0meeting at p0. Also, define a function fs: [0, s]→Rsuch that fs(t) CAT(0) AND CAT(-1) DIMENSIONS OF HYPERBOLIC GROUPS 19 is the distance from q0to a point a distance tfrom p0along the side of ∆0 between p0and r0. By the comparison axiom, we have f(t)≤fs(t) for every t∈[0, s]. Also by the comparison axiom, θ≤θsfor each s. Moreover, by the interpretation of Alexandrov angle as the “strong upper angle” (see Proposition I.1.16 of [11]), we have that lim s→0;s>0θs=θ. Now, suppose we are given a small ε > 0. Then there exists s∈(0, ²] such that θ≤θs< θ+ε. Since fsis a differentiable function with derivative −cos θs<−cos(θ+ε) (εsufficiently small) we may find a sufficiently small δsuch that f(δ)≤fs(δ)< f0−δcos(θ+ε). But this implies that for all sufficiently small ε > 0 there exists a δ > 0 such that −cos θ≤f(δ)−f0 δ<−cos(θ+ε). This establishes the Lemma. ¤ As in the main body of the proof of Theorem 9, we suppose that G= ha, b|bab2=a2iacts properly by isometries on a proper CAT(0) space X of geometric dimension two. We recall the notation introduced in the proof of Theorem 9. In particular, for a choice of point x∈X, we defined the map ϕx: Γ →X, where Γ denotes the Cayley graph of G(with respect to {a, b}). As before we write L= Lk(v, e K) and Σ = ΣxX, the space of directions at xin X, and we write pfor the point in Σ associated to a vertex pof Lvia the map ϕx. Recall that Lis a complete graph on 4 vertices. Since Xis CAT(0) the space of directions Σ is CAT(1). We recall that CAT(1) spaces are uniquely π-geodesic: if d(x, y)< π then there exists a unique geodesic in the space joining xto y. We also note that, by the dimension constraint on X, Σ is 1-dimensional (a CAT(1) metric R-graph). We note that, in a 1-dimensional CBA space, a path is locally godesic if and only if it is locally embedded. Lemma 12. Suppose that x∈Xis chosen so as to minimise the combined displacement f(x) = d(x, a(x))+d(x, b(x)). Let p1, p2, p3denote three distinct vertices of Land, for each i= 1,2,3, let φidenote the angle measured between piand pi+1 in Σ(indices taken mod 3), and suppose that each φi< π. Then either φ1+φ2+φ3>2πor the (unique) geodesic triangle in Σspanned by the vertices p1, p2, p3is a closed geodesic of length exactly 2π. Remark. Note that if φi=π, for some i, then the triangle inequality implies straightaway that φ1+φ2+φ3≥2π. Proof. Suppose that φ1+φ2+φ3≤2π. We show that the geodesic triangle spanned by p1, p2, p3in Σ is a closed geodesic of length exactly 2π. Since each φi< π, and since Σ is uniquely π-geodesic space, there is a unique 20 N. BRADY AND J. CRISP (but possibly degenerate) geodesic triangle ∆ spanned by p1, p2, p3. Either this triangle supports a simple closed circuit in Σ or it is in fact a (possibly degenerate) tripod. In the former case, since Σ is 1-dimensional and CAT(- 1), the simple circuit is a closed geodesic and must have length at least 2π. Thus ∆ is exactly a 2πclosed geodesic, as required. In the latter case we shall obtain a contradiction. Suppose then that the geodesic triangle ∆ is a tripod. We let mdenote the branch point of the tripod (or rather the median of p1, p2, p3– the unique point which lies on all three sides of the triangle ∆) and let θidenote the angle between piand m, for each i= 1,2,3. We have θ1+θ2+θ3= 1 2(φ1+φ2+φ3)≤π. We now consider the effect of moving the point xa very small distance in the direction m. For convenience we G-equivariantly subdivide all edges in the Cayley graph Γ. Let e1, e2, e3, e4denote the geodesic segments at xin Xwhich are the images under ϕxof the four half-edges of Γ which emanate from the vertex v. We suppose that the labels are such that ei determines the point piin Σ for each i= 1,2,3. We now allow xto move in the direction of mwhile fixing the other endpoints of the “half-edges” ei. Using Lemma 11 we may compute the derivative of `(ei) in the direction mat xto be simply −cos θifor each i= 1,2,3. On the other hand the derivative of `(e4) is at most 1 (in any direction). Thus it follows from Lemma 13 below that the sum of the lengths of the eistrictly decreases under a sufficiently small perturbation of the point x. (Note that since we suppose each φi< π we cannot have equality in Lemma 13). Performing this disturbance G-equivariantly, it is clear that the sum of the lengths of the new segments e0 iis an upper bound for f(x0) and hence that f(x0)< f(x) for a sufficiently small disturbance. But this contradicts the choice of x.¤ Lemma 13. Given real numbers θ1, θ2, θ3≥0such that θ1+θ2+θ3≤π, we have cos θ1+ cos θ2+ cos θ3≥1 with equality if and ony if θ1+θ2+θ3=πand θi= 0 for some i. Proof. The region of interest in R3is a right simplex R={(θ1, θ2, θ3) : θi≥0 and θ1+θ2+θ3≤π}. Write g(θ1, θ2, θ3) = cos θ1+ cos θ2+ cos θ3. We first consider the problem of minimisig the function gover the 2-simplex Rπ={(θ1, θ2, θ3)∈R: θ1+θ2+θ3=π}. Observe that g= 1 on the boundary of Rπ, namely when CAT(0) AND CAT(-1) DIMENSIONS OF HYPERBOLIC GROUPS 21 θ1+θ2+θ3=πand θi= 0 for some i. For, if θ1= 0 and θ2=π−θ3, for instance, then g(0, θ2, θ3) = 1 + cos(π−θ3) + cos θ3= 1. Consider now the possibility of local minima in the interior of Rπ. At such points the gradient of gis normal to Rπ. That is, ∇g=−(sin θ1,sin θ2,sin θ3) = λ(1,1,1) for some λ∈R. Thus sin θ1= sin θ2= sin θ3=−λ. But then one sees that θ1=θ2=θ3=π 3 (since if, for some i6=j, we had θi6=θjbut sin θi= sin θjwe would have θi+θj=πcontradicting the choice of point in the interior of Rπ). Now g(π 3,π 3,π 3) = 3 cos π 3=3 2>1. Therefore gis always strictly greater that 1 on the interior of Rπ. Finally, since gis strictly decreasing along rays from the origin (through R), we deduce that g > 1 at all points of R\Rπ.¤ Remark 14. In fact the barycentre of Rπis a local maximum of gover Rπ, as can be seen by looking at the Hessian matrix which is −cos(π 3)Inat the barycentre. More generally, an easy induction shows that g(θ1, .., θn) = n X i=1 cos θi≥n−2 whenever θiare non-negative and sum to at most π, with equality precisely on the 1-skeleton of the “level πsimplex” Rπ. 5. The hyperbolic versus CAT(0) problem: Theorem 2 Consider the following family of 2-dimensional groups: Gw,m =ha, b, t |abaa =bb, tamt−1=wi with m∈Z\{0}and wa positive word in a, b which contains at least one b. These are easily seen to be 2-dimensional, since they are HNN’s of the 2-dimensional group Gover infinite cyclic subgroups. In particular, they are all torsion free groups. That this family contains an infinite collection of hyperbolic groups which do not act properly isometrically on any proper CAT(0) space of dimension 2 follows by combining Proposition 16 and Proposition 17 below. More precisely, we obtain: Theorem 15. Let wto be any positive word which represents a primitive element of Gdifferent from a, and let m∈Zsuch that |m| ≥ L(w). Then Gw,m is a 2-dimensional hyperbolic group but does not act properly by isometries on any 2-dimensional CAT(0) space. 22 N. BRADY AND J. CRISP For example, one could take w=band |m| ≥ 3. (For other possible words wsee Remark 18 below). Finally, in Proposition 19, we see that when one chooses m=L(w), the resulting group is always a free-by-cyclic group. In the case of the previous example, the group Gb,3is isomorphic to F6o Z. These observations complete the proof of Theorem 2. Proposition 16. Let wdenote a positive word in the letters a, b which contains at least one b. If |m| ≥ L(w)then Gw,m admits no proper isometric action on a proper CAT(0) space of dimension 2. Proof. Suppose that Gw,m acts properly isometrically on a proper CAT(0) space Xof dimension 2. The group Gw,m is an HNN-extension of the group G=ha, b |abaa =bb iof Theorem 1. Thus Gacts properly isometrically on X(as a subgroup of Gw,m) and by Theorem 1 we have a copy of e Ktisometrically and G-equivariantly embedded in X. It follows that the translation lengths of the elements aand w(acting on X) may be measured in e Kt, and from Proposition 4 we have that l(w)/l(a)< L(w). However, the relation tamt−1=win Gw,m forces l(w)/l(a) = |m| ≥ L(w), a contradiction. ¤ Proposition 17. Let wdenote a positive word in the letters a, b which contains at least one b. If wrepresents a primitive element of the group G (and m6= 0) then Gw,m is a word hyperbolic group. Proof. The groups Gw,m are HNN extensions of the hyperbolic group G via an isomorphism identifying the cyclic subgroup hamiwith the cyclic subgroup hwi, so we can apply criterion (2) of Corollary 2.3 of the Bestvina- Feighn Combination Theorem [4]. Since the centralizer of any element in a torsion free hyperbolic group is always an infinite cyclic group, it follows that any primitive element wof G generates its own centralizer. Therefore hwiis malnormal in G, and so one of the conditions (a), (b) of criterion (2) in Corollary 2.3 holds. Note also that no non-trivial power of ais conjugate to a non-trivial power of w. For if this were the case then the ratio l(w)/l(a) would be constant over the full range of model spaces Ktfor G. However, the fact that l(w)/l(a) tends towards a strict upper bound (Proposition 8) shows that this is not the case. Thus, the set CC0(x) of criterion (2) of Corollary 2.3 is always finite (actually is always {1}). By criterion (2) of Corollary 2.3 of [4] and the results of the preceding two paragraphs, we conclude that the HNN extension Gw,m is torsion free hyperbolic whenever wis a (positive) primitive element of Gdifferent from a.¤ CAT(0) AND CAT(-1) DIMENSIONS OF HYPERBOLIC GROUPS 23 Remark 18. The only technical obstacle to applying the above propositions is knowing when a positive word wrepresents a primitive element of G. In many cases, however, we can give a geometric argument using geodesics in the universal cover of the regular hexagonal structure K0to prove primitivity. For suitable w, we observe that the piecewise geodesic which connects midpoints of successive edges in the bi-infinite edge-path determined by w is actually an axis for w.3This is the case if wis a positive word which is required not to contain either of the two positive, length 3 subwords (aba and baa) which form half of the hexagonal relator. If, moreover, whas odd wordlength then the axis is unique (wacts as a “glide reflection” along this axis). Since any root of wmust leave this axis invariant, it follows that wis primitive in Gif it is primitive in the free group F{a,b}(i.e: if it is not obviously a nontrivial power). Examples of such primitive elements w include b,ab2n(n≥1), ab2ab3etc. We note however that ab3= (a2b)2is not primitive. Many further elements may be seen to be primitive by variations on this argument. These include elements ab and a2b, as well as positive words which contain no subword aba or baa, other than the exception ab3 just mentioned. The next proposition shows that the examples afforded by Theorem 15 include infinitely many free-by-cyclic groups. Proposition 19. If m=L(w), then the group Gw,m is free-by-cyclic. Proof. Consider the group Gb,3=ha, b, t |abaa =bb, ta3t−1=bi for example. Recall that the original group Gadmits an epimorphism L:G→Z, where L(b) = 3L(a). Since the abelianization of the new relation still implies that L(b) = 3L(a), we can extend the circle-valued Morse function from the presentation 2-complex of Gto the presentation 2-complex for Gb,3 by mapping tonce around the circle, and extending “linearly” over the new 2-cell as shown in Figure 5. Ascending and descending links are trees (segments of length two each), so the space is homeomorphic to the total space of a graph of spaces where the underlying graph is again a one vertex circle, the vertex space is as shown on the left hand side of Figure 5, and the edge space is a graph which is homotopy equivalent to this, and maps are homotopy equivalences. Thus Gb,3is isomorphic to a semidirect product F6o Z. 3This reasoning enabled us earlier to compute exact translation lengths for the element a. 24 N. BRADY AND J. CRISP b3 t t b a a a v 2b PSfrag replacements a b t [b/3] [2b/3] v Figure 5. The Morse function on the extra 2-cell in the F6o Z group, and the level set through v. In the general case of the group Gw,L(w)the picture of the Morse function on the new 2-cell will be as in Figure 5, with a3replaced by aL(w)and with breplace by w. This has the effect of adding L(w) new edges to the level set Γ passing through v. 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The tangent cone of an Alexandrov space of curvature ≤K.Manuscripta Math. 86 (1995), 683–689. [20] D. Wise. Cubulating small-cancellation groups. Geom. Funct. Anal. , To appear. Dept. of Mathematics, University of Oklahoma, Norman, OK 73019 E-mail address:[email protected] Laboratoire de Topologie,, Universite de Bourgogne,, UMR 5584 du CNRS,, B. P. 47 870, 21087 Dijon, France E-mail address:[email protected]