Distortion of surface groups in Cat (0) free-by-cyclic groups

DISTORTION OF SURFACE GROUPS IN CAT(0) FREE-BY-CYCLIC GROUPS JOSH BARNARD AND NOEL BRADY1 Abstract. Given a non-positively curved 2-complex with a circlevalued Morse function satisfying some extra combinatorial conditions, we describe how to locally isometrically embed this in a larger nonpositively curved 2-complex with free-by-cyclic fundamental group. This embedding procedure is used to produce examples of CAT(0) free-by-cyclic groups that contain closed hyperbolic surface subgroups with polynomial distortion of arbitrary degree. We also produce examples of CAT(0) hyperbolic free-by-cyclic groups that contain closed hyperbolic surface subgroups that are exponentially distorted. Contents Introduction 1 1. Morse functions on 2-complexes 3 2. Quasi-convex surface subgroups 9 3. Polynomially distorted surface subgroups 12 4. Exponentially distorted surface subgroups 16 5. Questions 26 References 27 Introduction We introduce a technique for constructing non-positively curved 2-comple- xes with free-by-cyclic fundamental groups. As an application, we have a series of existence results concerning closed hyperbolic surface subgroups in free-by-cyclic groups. It is an open question as to whether every hyperbolic free-by-cyclic group contains a closed hyperbolic surface subgroup. Theorem 1 below can be Date: November 22, 2005. 1N. Brady was partially supported by NSF grant no. DMS-0124344. He also thanks the Centre de Recerca Matem`atica for their support and hospitality. 1 2 J. BARNARD AND N. BRADY viewed as a positive answer to the much more basic question: do there exist hyperbolic free-by-cyclic groups that contain closed hyperbolic surface subgroups? Theorem 1 (Quasi-convex surface subgroups).There exist hyperbolic freeby-cyclic groups containing quasi-convex closed hyperbolic surface subgroups. The following two theorems show that the distortion of closed hyperbolic surface subgroups in CAT(0) free-by-cyclic groups can be as varied as the distortion of free subgroups. We produce examples where the distortion is polynomial of arbitrary degree, or exponential. In fact, the proofs given here rely on the fact that there are CAT(0) free-by-cyclic groups whose free kernels have the appropriate distortion function. Theorem 2 (Polynomially distorted surface subgroups).For each integer n≥2, there exists a CAT(0) free-by-cyclic group containing the fundamental group of a closed hyperbolic surface that has polynomial distortion of degree n. In the exponential distortion case, we show that with quite a bit of extra care, one can require that the ambient free-by-cyclic group be hyperbolic. Theorem 3 (Exponentially distorted surface subgroups).There exist hyperbolic free-by-cyclic groups that contain exponentially distorted closed surface subgroups. The basic examples of hyperbolic groups that contain exponentially distorted, closed, hyperbolic surface subgroups are the fundamental groups of closed, hyperbolic 3-manifolds that fiber over the circle. Theorem 3 shows that one can find hyperbolic groups of cohomological dimension 2 containing exponentially distorted closed surface subgroups, and that, in particular, one can make the ambient hyperbolic group be free-by-cyclic. In all three theorems, the groups in question are fundamental groups of non-positively curved squared 2-complexes. The paper is organized as follows. In Theorem 1.8 of Section 1 we establish a criterion for when a non-positively curved 2-complex can be locally isometrically embedded in a larger non-positively curved 2-complex with free-by-cyclic fundamental group. The local isometric embedding ensures that the fundamental group of the original 2-complex embeds into the freeby-cyclic group. This is the basic way that we shall produce closed surface groups inside free-by-cyclic groups, but the technical difficulties in each section increase as we require more conditions either on the distortion of the surface group, or on the curvature of the ambient free-by-cyclic group, or both. DISTORTION OF SURFACE GROUPS 3 In Section 2 we apply this technique to a squared structure on a genus 2 surface to obtain a linearly distorted, or quasi-convex, subgroup of a hyperbolic free-by-cyclic group. The hyperbolicity is ensured by a careful application of the techniques used to prove Theorem 1.8 in Section 1. In Section 3 we take a CAT(0) amalgamation of (i) a free-by-cyclic group with polynomially distorted free subgroup and (ii) a closed surface group, amalgamating along the free subgroup. This produces a closed surface group with polynomial distortion inside a 2-dimensional CAT(0) group. We then embed this into a CAT(0) free-by-cyclic group by the methods of Theorem 1.8. In Section 4 we proceed as in Section 3, but this time starting with the amalgam of a surface group and a hyperbolic free-by-cyclic group, and proceed carefully as in Section 2 in order to end with an ambient hyperbolic free-by-cyclic group. Finally, in Section 5 we pose questions that are related to these ideas and constructions. Noel Brady thanks Martin Bridson and Jens Harlander for conversations about CAT(0) LOT groups and free-by-cyclic constructions. 1. Morse functions on 2-complexes In this section we recall a Morse theory way of recognizing that a 2- complex is aspherical and has free-by-cyclic fundamental group. We then describe a combinatorial condition on non-positively curved 2-complexes (the existence of f-link filling graphs) that ensures that they locally isometrically embed into non-positively curved 2-complexes with free-by-cyclic fundamental groups. The material in this section is key to understanding the constructions in the subsequent sections of the paper. The first two definitions below are from [1]. Definition 1.1 (Affine cell complexes).A finite-dimensional cell-complex Xis said to be an affine cell-complex if it is equipped with the following structure. An integer m≥dim(X) is given, and for each cell eof Xwe are given a convex polyhedral cell Ce⊂Rmand a characteristic function χe:Ce→esuch that the restriction of χeto any face of Ceis a characteristic function of another cell, possibly precomposed by a partial affine homeomorphism (=restriction of an affine homeomorphism) of Rm. Definition 1.2 (Morse function).A map f:X→Rdefined on an affine cell complex Xis a Morse function if •for every cell eof X,fχe:Ce→Rextends to an affine map Rm→R, and fχeis constant only when dim e= 0, and •the image of the 0-skeleton is discrete in R. 4 J. BARNARD AND N. BRADY Definition 1.3 (Circle-valued Morse function).A circle-valued Morse function on an affine cell complex Xis a cellular map f:X→S1, with the property that flifts to a Morse function between universal covers. By composing with a covering map S1→S1if necessary, we will assume that all circle-valued Morse functions map to a circle with a single vertex. Definition 1.4 (Ascending and descending links).Suppose Xis an affine complex and f:X→S1is a circle-valued Morse function. Choose an orientation of S1, which lifts to one of R, and lift fto a map of universal covers e f:e X→R. Let v∈X(0), and note that the link of vin Xis naturally isomorphic to the link of any lift ˜vof vin e X. We say that a cell ˜e⊂e Xcontributes to the ascending (respectively descending) link of ˜v if ˜v∈˜eand if e f|˜eachieves its minimum (respectively maximum) value at ˜v. The ascending (respectively descending) link of vis then defined to be the subset of Lk(v, X) naturally identified with the ascending (respectively descending) link of ˜v. Note that in the case Xis a 2-complex, all the ascending, descending and entire links will be graphs. Henceforth, all cell complexes will be 2-complexes. The next proposition gives a local way of telling that a 2-complex is aspherical and has free-by- cyclic fundamental group. Proposition 1.5 (Free-by-cyclic).If f:X→S1is a circle-valued Morse function on the 2-complex Xall of whose ascending and descending links are trees, then Xis aspherical, and π1(X)is free-by-cyclic. Remark 1.6. This is basically a result of J. Howie [5]. In [5] it is proved that if all the ascending links (or all the descending links) of a given LOT 2- complex are trees, then the 2-complex is aspherical. Also, if both ascending and descending links are trees, then the LOT 2-complex has free-by-cyclic fundamental group. The current proposition says that if all of both the ascending and the descending links of a general affine 2-complex are trees, then the 2-complex is aspherical and has free-by-cyclic fundamental group. Note that if just the ascending links are trees and the descending links are not, then the aspherical 2-complex need not have free-by-cyclic fundamental group. As a simple example, take the group with five generators tand ai (i= 0,...,3) and four conjugation relations aiai+1a−1 i=t(imod 4). In any map to Z, all five generators must have the same image, so we can assume this common image is a generator of Z. The corresponding circlevalued Morse function takes the aiand the tloops once around a target circle. Its ascending link is a tree, but the descending link is the disjoint union of a single vertex and a circle. The kernel of the map to Zis not finitely generated. DISTORTION OF SURFACE GROUPS 5 Proof of Proposition 1.5. Look at generic fpoint preimages in X. Since X has an affine structure and f:X→S1is affine, the point preimages are affine graphs in X. We may then view Xas the total space of a graph of spaces: the underlying space is the target circle, the edge space is the generic point preimage graph, and the vertex space is the S1-vertex preimage. The maps from an edge space to the adjacent vertex spaces collapse either the ascending or the descending subgraphs of the corresponding link. These are trees by hypothesis, so these maps are homotopy equivalences. It follows that Xis homotopy equivalent to a graph bundle over S1, and therefore is aspherical and has free-by-cyclic fundamental group.  The next definition is a little technical, so we preface it with a discussion of the intuition behind it. We are given a circle-valued Morse function on a locally CAT(0) 2-complex X, with the property that all ascending and descending links are forests. We would like to be able to attach 1-cells and 2-cells to Xto obtain a bigger locally CAT(0) 2-complex in which Xlocally isometrically embeds. Furthermore, we want to extend the Morse function over the resulting 2-complex, so that all ascending and descending links are now trees. The description of the new 2-complex will be as a graph of spaces, whose underlying graph is a circle with one vertex, and one 1-cell. The vertex space is the original 2–complex X, and the edge space is a graph Γ. That is, the new 2–complex is obtained from Xby attaching a “handle” of the form Γ ×[0,1]. Condition (1) in Definition 1.7 is there to ensure that the geometry works out, namely, that the new 2-complex is locally CAT(0), and that the inclusion map of Xis a local isometric embedding. Condition (2) in Definition 1.7 is there to ensure that all the ascending and descending forests get connected up precisely into trees. This intuition will be made precise in the proof of Theorem 1.8. Definition 1.7 (f-link filling graphs).Suppose f:X→S1is a circlevalued Morse function on a locally CAT(0) 2-complex X. An f-link filling graph is a (not necessarily connected) graph Γ satisfying the following: (1) there are two combinatorial immersions ϕ1, ϕ2: Γ →X(1) that are local isometric embeddings, (2) the maps (f◦ϕi)∗:π1(Γ) →π1(S1) agree, and (a) for every vertex v∈Xthe induced map on the direct sum of reduced homology groups M w∈Γ(0) ϕ1(w)=ve H0(LK↑(w, Γ); Z)→e H0(Lk↑(v, X); Z) is an isomorphism, and 6 J. BARNARD AND N. BRADY (b) for every vertex v∈Xthe induced map on the direct sum of reduced homology groups M w∈Γ(0) ϕ2(w)=ve H0(LK↓(w, Γ); Z)→e H0(Lk↓(v, X); Z) is an isomorphism. Note that in part 2(a) the ascending links of Γ are with respect to the Morse function f◦ϕ1, and in part 2(b) the descending links are with respect to the Morse function f◦ϕ2. We often refer explicitly to the immersions and write that ϕ1, ϕ2: Γ →Xis f-link filling. As indicated before the definition, the main application we have in mind for an f-link filling graph Γ is when the Morse function f:X→S1has forests as its ascending and descending links. Then the new 2-complex obtained from Xby attaching Γ ×[0,1] via the map ϕ1at the Γ ×0 end, and the map ϕ2at the Γ ×1 end, will admit a circle-valued Morse function with trees as ascending and descending links. Here are the details. Theorem 1.8 (Free-by-cyclic completions).Let Xbe a CAT(0) 2-complex Xwith a circle-valued Morse function f:X→S1whose ascending and descending links are all forests. If ϕ1, ϕ2: Γ →Xis an f-link filling graph, then the 2-complex, C(X), obtained from Xby adjoining Γ×[0,1] via ϕ1 on Γ×0, and ϕ2on Γ×1satisfies: (1) the complex C(X)is locally CAT(0) and the inclusion X→C(X) is a local isometric embedding, (2) the Morse function f:X→S1extends to a Morse function b f: C(X)→S1all of whose ascending and descending links are trees. In particular, π1(X)is a subgroup of π1(C(X)), and π1(C(X)) is a CAT(0), free-by-cyclic group. Proof of Theorem 1.8. The 2-complex C(X) is obtained from the nonpositively curved 2-complex Xby attaching Γ×[0,1] via the local isometric embeddings ϕi. Here Γ×[0,1] is given the product metric, and the complex C(X) gets the induced length metric. Note that we have just attached 1- cells (in the form of {w} × [0,1] where wis a vertex of Γ) and 2-cells (in the form of e×[0,1] where eis an edge of Γ) to X. Thus, C(X) and Xhave the same vertex set. Proof of part (1). We show that C(X) is non-positively curved, and that the inclusion X⊂C(X) is a local isometric embedding. This follows from the standard theory [4], but we give a proof here, as the local picture that is developed below is used in the proof of part (2). DISTORTION OF SURFACE GROUPS 7 For non-positive curvature, we must verify that Lk(v, C(X)) is ‘large’ for every vertex v∈C(X); i.e., we check that all cycles in Lk(v, C(X)) have length at least 2π. From the previous paragraph, we know that v∈X, and the non-positive curvature of Xensures that Lk(v, X) is large. Thus we have to describe the effect on Lk(v, X) of attaching Γ ×[0,1] via the maps ϕi. Let us consider the ϕ1attaching map; the analysis of ϕ2is completely analogous. If w∈Γ(0) then Lk((w, 0),Γ×[0,1]) is simply an orthogonal cone on Lk(w, Γ). The cone point corresponds to the edge w×[0,1] at (w, 0), and the adjective “orthogonal” just means that the lengths of the edges are all π/2. This latter fact holds since we’ve taken the product metric on Γ ×[0,1]. Since ϕ1is a local isometric embedding we know that the ϕ1-induced image of Lk(w, Γ) in Lk(v, X), denoted by (ϕ1)∗(Lk(w, Γ)), is convex. Specifically, (ϕ1)∗(Lk(w, Γ)) is a discrete set of points (in 1-1 correspondence with Lk(w, Γ)) that is π-separated. Thus, the local effect of attaching a neighborhood of (w, 0) in Γ×[0,1] via ϕ1is to attach an orthogonal cone on Lk(w, Γ) to Lk(v, X) by identifing Lk(w, Γ) with the π-separated subset (ϕ1)∗(Lk(w, Γ)) of Lk(v, X). This new graph is still large, and Lk(v, X) is a convex subgraph (in the CAT(1) sense). We continue to attach orthogonal cones, one for every w∈Γ with ϕ1(w) = vand one for every w∈Γ with ϕ2(w) = v. Thus the final result, Lk(v, C(X)), is large, and Lk(v, X) is a convex subgraph. This shows that C(X) is non-positively curved, and that X⊂C(X) is a local isometric embedding. Note that we have needed only the hypothesis that the ϕiare local isometric embeddings for this part. Proof of part (2). The requirement that (f◦ϕ1)∗= (f◦ϕ2)∗ensures that the Morse function f:X→S1extends to a Morse function b f:C(X)→S1. We can explicitly construct this extension as follows. For any edge e⊂Γ, we let N1(e) denote the (signed) number of times that (f◦ϕ1)(e) winds around S1, and similarly define N2(e). Choose a vertex w0∈Γ and an integer t0so that t0>X e⊂Γ |N2(e)−N1(e)|. We map both end points of the new 1-cell {w0} × [0,1] to the vertex of the circle and send the [0,1] interval around the circle t0times in the positive direction. Now suppose w1is a vertex of Γ adjacent to w0via the edge e1. We map the new 1-cell {w1}×[0,1] around the circle monotonically t1=t0+ (N2(e1)−N1(e)) times. Note that t1is always positive by choice of t0, and so this 1-cell gets mapped positively around the target circle. Continuing in this fashion, suppose we have some sequence of edges e1,...,enwith ei joining wi−1to wiand with wn=w0. We need to check that t0=tn. Set γ∈π1(Γ) to be that element represented by the concatenation of the edges 8 J. BARNARD AND N. BRADY e1,...,en, and note that tn−t0= n X i=1 N2(ei)−N1(ei) = n X i=1 N2(ei)− n X i=1 N1(ei) = (f◦ϕ2)∗(γ)−(f◦ϕ1)∗(γ) = 0. It follows that we may extend consistently across the edges of C(X). Note also that this construction implies that the boundaries of the 2-cells {e} × [0,1] are mapped null-homotopically into S1, so that we may extend “linearly” across these new 2-cells, completing the extension of fto b f. We have to argue that the new ascending and descending links are trees. We give the argument for ↑-links. The argument for ↓-links is analogous: just interchange the roles of ϕ1and ϕ2in the argument below, and use hypothesis 2(b) in place of hypothesis 2(a). Let v∈C(0)(X). Since b fis an extension of f, we have that Lk↑(v, X)⊂ Lk↑(v, C(X)). Here the first ↑-link is with respect to f, and the second ↑-link is with respect to b f. By the definition of the extension, any lift of b fto the universal covers will attain a maximum on an edge {w} × [0,1] at the (w, 1) endpoint, and will attain a minimum at the (w, 0) endpoint. Thus there are no contributions to Lk↑(v, C(X)) by the (ϕ2)∗-attached orthogonal cones. Let w∈Γ be such that ϕ1(w) = v. In the proof of part (1) we saw that this contributed an orthogonal cone on (ϕ1)∗(Lk(w, Γ)) to Lk(v, C(X)). Of this, the contribution to Lk↑(v, C(X)) is simply the cone on the subset (ϕ1)∗(Lk↑(w, Γ)). Here we think of ϕ1as defining a Morse function on Γ by composition with f. To complete the proof, we claim that by coning off each of the sets (ϕ1)∗(Lk↑(w, Γ)) as wranges over all those vertices of Γ that are in ϕ−1 1(v), the components of the forest Lk↑(v, X) are connected to produce a tree. For this we attach one C(Lk↑(w, Γ)) at a time to Lk↑(v, X) and apply the following lemma. The precise argument is continued after the lemma. Lemma 1.9. If f:P→Qis a map of spaces so that f∗:˜ H0(P)→˜ H0(Q) is injective, and ˜ H1(Q) = 0, then (1) ˜ H1(C(P)∪fQ) = 0, and (2) ˜ H0(C(P)∪fQ) = ˜ H0(Q)/f∗(˜ H0(P)) where C(P)denotes the cone on P. Proof of Lemma 1.9. This is just reduced Mayer-Vietoris. We have →˜ H1(C(P)) ⊕˜ H1(Q)→˜ H1(C(P)∪fQ)→˜ H0(C(P)∩Q)→ ˜ H0(C(P)) ⊕˜ H0(Q)→˜ H0(C(P)∪fQ)→0 DISTORTION OF SURFACE GROUPS 9 Since C(P) is contractible, all its reduced homologies vanish. By hypothesis, the reduced 1st homology of Qis trivial. Thus the first term on the left is trivial. The ˜ H0(C(P)∩Q) term is just a copy of ˜ H0(P) and so injects (via f∗) into ˜ H0(Q) by hypothesis. Since this map injects, its kernel is trivial, and so we conclude that ˜ H1(C(P)∪fQ) is trivial. This proves (1). The sequence now becomes 0→˜ H0(P)→˜ H0(Q)→˜ H0(C(P)∪fQ)→0 which gives us part (2).  Let {w1,...,wn}be the set of vertices of Γ that are mapped to vby ϕ1. Start with P=Lk↑(w1,Γ) and Q=Lk↑(v, X). Use Lemma 1.9 to determine the effect of attaching a cone on Lk↑(w1,Γ) to Q=Lk↑(v, X). Continue to attach cones on the Lk(wi,Γ), using Lemma 1.9 at each stage to determine that the resulting graphs all have trivial first homology, and that their reduced 0-homology groups are quotients of ˜ H0(Lk↑(v, X)) by sums ⊕1≤j≤if∗(˜ H0(Lk↑(wj,Γ))). The isomorphism hypothesis 2(a) in the definition of f-link filling ensures that we can continue applying Lemma 1.9 until the last cone has been attached, at which point the reduced 0-homology group will be trivial. This concludes the proof of Theorem 1.8.  2. Quasi-convex surface subgroups Proof of Theorem 1. We start by taking the 2-complex Xof Theorem 1.8 to be a closed hyperbolic surface. For example we can take Xto be the surface of genus 2 obtained from the top 3 squares of Figure 1 by performing the indicated edge-pair identifications. The circle-valued Morse function is defined to take each oriented edge once around the target circle, and is extended linearly over the 2-cells of the surface X. We now perform the completion outlined in Theorem 1.8 so that the universal cover of the resulting complex C(X) does not contain any isometrically embedded flat planes. This will ensure that the resulting group is hyperbolic. This is done by taking the graph Γ of Theorem 1.8 to be the disjoint union of two copies of the wedge of two circles. The maps ϕiof Theorem 1.8 are defined as follows: ϕ1maps the first bouquet of two circles to the bouquet in X(1) labeled by the ordered pair (x, y), and the second bouquet to the bouquet in X(1) labeled by (a, b), while the map ϕ2takes the first bouquet to the bouquet in X(1) labeled by (z, x) and takes the second bouquet to the bouquet in X(1) labeled by (b, z). This can be easily read off from the bottom 4 squares in Figure 1; identifying the vertical sand tedges gives the complex Γ×[0,1], and the labels on the top and bottom bouquets define the 16 J. BARNARD AND N. BRADY β1 β2 . . . . . . βn+1 b αm xi−1 xj yi yk βi−1 βi+1 Figure 7. The graph Γ in the application of Theorem 1.8. have polynomial distortion of degree nin the CAT(0) free-by-cyclic group π1(C(X)). We claim that the graph Γ shown in Figure 7 works for any choice of i, j,kand mwith i6=k, j + 1. We define a map ϕ: Γ →X(1) according to the labels and orientations on the edges in Figure 7, and set ϕ1=ϕ2=ϕin constructing C(X). It is easy to check that ϕis a local isometric embedding. We now show that the resulting ascending and descending links are trees. We let t1,...,t4denote the edges of Γ ×[0,1] reading from left to right in Figure 7. Each ascending and descending link in Figures 5 and 6 receives two new vertices, labeled and attached as follows: To Lk↑(v1, X) we add a new vertex t− 1, joined by an edge to x− i−1, and a new vertex t− 2, joined to x− j. To Lk↓(v1, X) we add a new vertex t+ 1, joined to x+ j, and a new vertex t+ 2, joined to x+ i−1,bα+ m, and all the β+except β+ i. To Lk↑(v2, X) we add a new vertex t− 4, joined to y− k, and a new vertex t− 3, joined to y− i,bα− m, and all the β−except β− i. To Lk↓(v2, X) we add a new vertex t+ 3, joined to y+ k, and a new vertex t+ 4, joined to y+ i. It is easy to check that the new ascending and descending links are all trees.  4. Exponentially distorted surface subgroups In this section we prove Theorem 3. We do this by constructing a nonpositively curved squared complex with hyperbolic, free-by-cyclic fundamental group, that contains an exponentially distorted closed surface group. The outline of the construction is similar to that of the previous section. DISTORTION OF SURFACE GROUPS 17 (1) We begin with a non-positively curved squared complex with hyperbolic F7⋊ Z fundamental group. Note that the fiber F7is exponentially distorted in F7⋊ Z. (2) As in the previous example, we next glue a double cover Kof this complex to a suitably chosen closed surface to obtain X. Note that the surface group injects into π1(X) by the standard theory of free products with amalgamation. Moreover, we choose the gluing, essentially identifying the F7fiber with a highly convex F7in the surface group, so that the distortion of the surface group in π1(X) is exponential. (3) Finally, we apply Theorem 1.8 to obtain a non-positively curved 2-complex C(X) so that X→C(X) is a local isometric embedding. As in the proof of Theorem 1, the completion is done with care to ensure that π1(C(X)) is hyperbolic. In our construction π1(C(X)) is isomorphic to F65 ⋊ Z. Here are the details. Proof of Theorem 3. (1): The free-by-cyclic group, and the complex K. a0 a1 a1 a2 a2 a3 a3 a4 a4 a5 a5 a6 a6 a7 a7 a+ 0 a+ 1 a+ 2 a+ 3 a+ 4 a+ 5 a+ 6 a+ 7 a− 0 a− 1 a− 2 a− 3 a− 4 a− 5 a− 6 a− 7 Figure 8. LOT presentation and link of vertex in presentation 2-complex. We start with the LOT group corresponding to the labeled oriented tree on the left hand side of Figure 8. The link of the vertex in the corresponding presentation 2-complex is shown on the right side of Figure 8. From this we see that the LOT group is CAT(0). As in the proof of Theorem 1, 18 J. BARNARD AND N. BRADY if Pis an isometrically embedded flat plane in the universal cover of the presentation 2-complex, and if w∈σ⊂Pis a vertex of a square σin P, then the segment Lk(w, σ) is contained in the embedded length 4 circuit, Lk(w, P), inside the link of win the universal cover. Again, links in the universal cover are isomorphic to the link of the single vertex vin the LOT complex. The only circuits of length 4 in this link contain a− 0and a+ 0as opposite vertices. Thus each 2-cell in Pcontains at least two adjacent corners of the form Xa± 0, where X∈ {a− 1, a− 2.a− 3, a− 4, a+ 5, a− 6, a− 7}, and thus two corners of the form XY , where X∈ {a+ 1, a+ 2, a+ 3, a+ 4, a− 5, a+ 6, a+ 7}. Such corners correspond to segments not contained in any embedded circuits of length four in the ambient link. Therefore, the universal cover of the presentation 2-complex does not contain any flat planes, and so the LOT group is hyperbolic. Essentially, the largest pieces of flat planes that exists in the universal cover are strips of width 2 squares which are centered on translates of the a0-axis. There is a circle-valued Morse function on the presentation 2-complex of this group, defined by mapping each directed edge once around the target circle. The ascending and descending links are trees; they are the sublinks of the link in Figure 8 spanned by the negative and by the positive vertices respectively. Proposition 1.5 implies that the LOT group is free-by-cyclic, and an examination of the rank of the vertex preimage shows that it is in fact F7⋊Z. Finally, since the ambient group is hyperbolic, the F7subgroup is exponentially distorted in the LOT group. The LOT group can be mapped onto Z2by sending each aito the nontrivial element of Z2. The complex Kis the double cover of the presentation 2-complex corresponding to the kernel of this epimorphism. Its fundamental group is also hyperbolic of the form F7⋊ Z where the F7subgroup is exponentially distorted. The 1-skeleton of Khas two vertices, v1and v2, and the subgraph Θ ⊂K(1) consisting of directed edges a0,...,a7from v1 to v2has fundamental group isomorphic to this F7. We let ba0,...,ba7denote the corresponding edges directed from v2to v1. (2): The hyperbolic surface Σ, and the complex X= Σ ∪ΘK. a0 a0a1a2a3a4a5a6a7 x0 x0x1 x1x2 x2x3 x3x4 x4x5 x5x6 x6x7 x7 y0 y0y1 y1y2 y2y3 y3y4 y4y5 y5y6 y6y7 y7 b0b1b2b3b4b5b6b7 Figure 9. The non-positively curved, squared surface Σ. DISTORTION OF SURFACE GROUPS 19 Identifying the a0edge pairs, the xiedge pairs, and the yiedge pairs in Figure 9 produces a closed hyperbolic surface Σ with two vertices. Let v1be the vertex where the yiloops are based, and v2be the vertex where the xiloops are based. The surface Σ contains a copy of the graph Θ. Identifying the surface Σ with the complex Kalong the copies of Θ gives a non-positively curved squared complex X= Σ ∪ΘKwith hyperbolic fundamental group. To verify this, it is sufficient to check that Lk(v1, X) and Lk(v2, X) are both large, and then to check that e Xdoes not contain any isometrically embedded flat planes. To this end, note that because the links Lk(vi, K) are large, any short loop in Lk(vi, X) must contain a segment of Lk(vi,Σ). But each Lk(vi,Σ) is a circle with combinatorial length 32 with the distance between any two a± iat least four (see Figure 10). Thus the links Lk(vi, X) are large. Moreover, the only circuits of length four come from those in K. As the original presentation 2-complex of the LOT group (which is double covered by K) has no flat planes in its universal cover, it follows that π1(X) is hyperbolic. a− ia+ i b+ i b− i y+ i y− ix+ i x− i b+ i−1a− i+1 a+ i+1 x+ i−1 Figure 10. Pieces of the surface links Lk(v1,Σ) and Lk(v2,Σ). Note that the surface group π1(Σ) is a subgroup of the hyperbolic group π1(X), since π1(X) is obtained from π1(Σ) via free product with amalgamation. Note also that this surface subgroup is exponentially distorted in π1(X), since the amalgamating free subgroup π1(Θ) is quasi-convex in π1(Σ) and is exponentially distorted in π1(K) and hence in π1(X). All that remains is to apply Theorem 1.8, choosing Γ carefully so that π1(C(X)) is hyperbolic. (3): The completion of Xto C(X). Figure 11, without edge labels and orientations, shows Γ ×[0,1], with Γ×0 at the bottom and Γ ×1 at the top. The maps ϕ1and ϕ2are defined as indicated by the edge labels and orientations on Γ ×0 and Γ ×1, respectively. The complex C(X) is the union of Xwith Γ ×[0,1] glued together using the maps ϕ1and ϕ2. Hereinafter, all subscripts are to be interpreted modulo 8. There are several conditions that must be satisfied by the various unspecified subscripts involved in the definition of the maps ϕi. For the present, we note that there are precisely four choices of triple (z, p, q) that work: (3,6,7), (4,7,3), (6,2,1), and (7,2,6). The conditions imposed 20 J. BARNARD AND N. BRADY on the choices of subscripts for the xand yedges are more easily met. In particular, the construction will work as long as j, k, n, m are distinct and dfferent both from zand from z−1. bz bz bz+1 bz−1 bz−1 b0 b0 b7 b7 bz−2 . . . . . . . . . . . . b ap b aq xm xn xj xk ym yn yj yk az az v1 v1 v1 v1 v1 v1 v2 v2 v2 v2v2 v2 r1r2s1s2t1t2 Figure 11. The complex Γ ×[0,1]. One may now check that these maps ϕ1and ϕ2are f-link filling. It is also possible to check directly that the conclusions of Theorem 1.8 hold for C(X). We proceed as follows: describe the extension of the Morse function, show that the new ascending and descending links are all trees, argue that the complex C(X) has large links, and that the subcomplex X locally isometrically embeds, and finally argue that π1(C(X)) is hyperbolic. Morse function extension. Here is an explicit description of a Morse function b f:C(X)→S1which extends f. We need to extend the function over the 2-cells of Γ ×[0,1]. This will be by the usual “linear” extension, once we have defined the Morse function on the 1-cells of Γ×[0,1]. We have already defined the function on the 1-cells in Γ×0 and Γ×1 since these are identified with 1-cells of X, and fis already defined on X. So we only have to define the function b fon the six 1-cells of the form w×[0,1], where wis a vertex DISTORTION OF SURFACE GROUPS 21 of Γ. Label these edges as r1, r2, s1, s2, t1, t2, as indicated in Figure 11, and orient each edge from the 0 end to the 1 end. The map b fsends each ti edge once around the target circle, sends s2and each rithree times around the target circle, and sends s1five times around the target circle. Thus from the perspective of the Morse function b f, the 2-cells of Γ ×[0,1] are parallelograms and trapezoids, as shown in Figure 12. The unlabeled edges in the center trapezoid of Figure 12 range over {bz+1,...,b7,baq, b0,...,bz−1} for the upper edge, and over {bz,...,b7,bap, b0,...,bz−2}for the lower edge. r1 r1 r2 r2 r2 s1 s1 s2 s2 t1 t1 t1 t2 t2 bz bz−1 yj yk xnxm ym yn xj xk az az Figure 12. The Morse function b fon the new 2-cells of C(X). Ascending and descending links. The ascending and descending links with respect to b fare described as follows. Start with the ascending and descending links of v1and v2in the 2-complex X. These are drawn in Figure 13 for v1and in Figure 14 for v2. To form the corresponding ascending and descending links in C(X), we add three new vertices to each of these links, labeled and attached as follows: In Lk↑(v1, C(X)), the three new vertices are: r− 1, which is joined by a new edge to y− m;r− 2, which is joined by an edge to y− n; and s− 2, which is joined by an edge to a− z. In Lk↓(v1, C(X)), the three new vertices are: s+ 1, which is joined to both ba+ qand all the b+ i;t+ 1, which is joined to y+ k; and t+ 2, which is joined to y+ j. In Lk↑(v2, C(X)), the new vertices are: t− 2, joined to x− k;t− 1, joined to x− j; and s− 1, joined to both ba− pand all the b− i. In Lk↓(v2, C(X)), the new vertices are: r+ 2, joined to x+ m;r+ 1, joined to x+ n; and s+ 2, joined to a+ z. Clearly all ascending and descending links in C(X) are trees. Large links and local isometric embedding. Now we verify that the links of the vertices viin C(X) are large. We have already seen that the links of the viin Xare large. From the proof of Theorem 1.8, we know that we 22 J. BARNARD AND N. BRADY a− 0 a− 1 a− 2 a− 3 a− 4 a− 5 a− 6 a− 7 b a+ 0 b a+ 1 b a+ 2 b a+ 3 b a+ 4 b a+ 5 b a+ 6 b a+ 7y− 0 y− 1 y− 2 y− 3 y− 4 y− 5 y− 6 y− 7 y+ 0 y+ 1 y+ 2 y+ 3 y+ 4 y+ 5 y+ 6 y+ 7 b+ 0 b+ 1 b+ 2 b+ 3 b+ 4 b+ 5 b+ 6 b+ 7 Figure 13. Descending and ascending links of v1in X. only have to check that the maps ϕiare local isometric embeddings. To check that ϕ1is a local isometric embedding at v1we need to verify that the following two subsets of Lk(v1, X) are π-separated: {y− m, y+ n},{y+ m, y− n, b+ z−1},{a− z, b+ 0,...,b+ z−2, b+ z,...,b+ 7,ba+ p}, and to check that ϕ1is a local isometric embedding at v2we need to verify that the following subsets of Lk(v2, X) are π-separated: {b− 0,...,b− 7,ba− p},{a+ z, x− j, x+ k},{x+ j, x− k}. Likewise, to see that ϕ2is a local isometric embedding at v1we need to verify that the following subset of Lk(v1, X) is π-separated: {b+ 0,...,b+ 7,ba+ q},{a− z, y− j, y+ k},{y+ j, y− k}, and to check that ϕ2is a local isometric embedding at v2we need to verify that the following two subsets of Lk(v2, X) are π-separated: {x+ n, x− m},{x− n, x+ m, b− z},{a+ z, b− 0,...,b− z−1, b− z+1,...,b− 7,ba− q}. As long as we choose z6=j6=k6=z−1 and z6=n6=m6=z−1, and choose pand qso that neither (ba+ p, a− z) nor (ba− q, a+ z) are adjacent pairs in the appropriate Lk(vi, K), these sets will be π-separated in the links both DISTORTION OF SURFACE GROUPS 23 a+ 0 a+ 1 a+ 2 a+ 3 a+ 4 a+ 5 a+ 6 a+ 7 b a− 0 b a− 1 b a− 2 b a− 3 b a− 4 b a− 5 b a− 6 b a− 7 x− 0x− 1x− 2x− 3x− 4x− 5x− 6x− 7 x+ 0 x+ 1 x+ 2 x+ 3 x+ 4 x+ 5 x+ 6 x+ 7 b− 0b− 1b− 2b− 3b− 4b− 5b− 6b− 7 Figure 14. Ascending and descending links of v2in X. of Kand of Σ. It follows that they are π-separated also in the appropriate links of X. The fundamental group of C(X)is hyperbolic. It remains only to show that π1(C(X)) is hyperbolic. Equivalently, we have to check that the universal cover of C(X) does not contain any isometrically embedded flat planes. For this it is helpful to have a more complete description of the links in C(X). Each Lk(vi, C(X)) is the union of Lk(vi, K), Lk(vi,Σ), and edges coming from the corners of squares in Γ ×[0,1]. Figure 8 shows the links in K(replacing a+ iwith ba+ iin the link of v1, while a− ibecomes ba− iin the link of v2). Figure 10 shows Lk(vi,Σ), while the new portions coming from Γ×[0,1] are shown in Figure 15, where the unlabeled vertices are, on the left, all the b+ iexcept b+ z−1, and on the right, all the b− iexcept b− z. We first show that no squares lifted from Σ can be contained in any flat plane P. Suppose σis a square in Pcovering a square in Σ. The square in Σ covered by σhas either two xisides or two yisides. Consider first the case that this square has two xisides, and let wbe the corner of σshared by the lifts of these two sides in σ. Thus Lk(w, P) is a circuit of length four with vertices labeled x+ ix− iAB for some labels Aand B. We will show that there are no choices of Aand Bthat can complete this circuit. Figure 16 shows the possible choices for Aand B, as they depend on i. We need to show that none of the vertices on the far right in Figure 16 is adjacent to 24 J. BARNARD AND N. BRADY r− 1r+ 1 r− 2r+ 2 s− 1 s+ 1 s− 2s+ 2 t− 1 t+ 1 t− 2 t+ 2 x− j x+ j x− k x+ k x− n x+ n x− m x+ m y− j y+ j y− k y− k y− n y+ n y− m y+ m a− za+ z b− z b+ z−1 b a− p b a+ p b a− q b a+ q Figure 15. The portions of Lk(v1, C(X)) and Lk(v2, C(X)) contributed to by Γ ×[0,1]. x+ iin Lk(v2, C(X)). From Figures 10 and 15 we see that x+ iis not adjacent to a+ i,b− z, or any other x+ i′. We can rule out a+ zby choosing j6=z−1. x+ ix− i r+ 2 r+ 1 b− i t− 2 t− 1 b− z x+ m x+ n a+ i x+ j x+ k a+ z (i=n) (i=m) (i=k) (i=j)   Figure 16. Finding a circuit around a vertex in P. DISTORTION OF SURFACE GROUPS 25 The case that σcontains two sides labeled yiis similar. If the purported length four circuit in Pis of the form y− iy+ iAB, one finds that possibilities for Bare other y− i′,b+ z−1, or a− i+1 (none of which is adjacent to y− i), or possibly a− z, in the case i=k. Thus by choosing k6=z, we ensure there are no length four circuits of this form. This rules out the presence of squares lifted from Σ in any purported flat plane P. Now consider the possibility that Pcontains squares lifted from Kand from Γ ×[0,1]. In particular, there is some edge of Palong which a square from Kmeets a square from Γ×[0,1]. This edge must be a lift either of az, bap, or baq. In any case, this edge jonis lifts of v1and of v2. Let vbe the lift of v1on this edge. Then Lk(v, P) is a circuit of length four in Lk(v1, C(X)), with vertices labeled, say XY ZW, where X∈ {a− z,ba+ p,ba+ q}. Note that there must be some other edge incident to vlying between a square lifted from Kand a square lifted from Γ ×[0,1]. In particular, one of Y,Z, or W is also labeled a− z,ba+ p, or ba− q. If we choose z,p, and qso that these vertices all have combinatorial distance at least three in Lk(v1, C(X)), it will follow that Pcannot contain squares lifted from Kand from Γ ×[0,1]. There are precisely four choices for the triple (z, p, q) satisfying this, and all previously imposed, conditions. They are (3,6,7), (4,7,3), (6,2,1), and (7,2,6). Because π1(K) is hyperbolic, no flat plane can consist entirely of squares lifted from K. Thus with any of the four allowed choices for (z, p, q), we find that any flat plane Pconsists entirely of squares lifted from Γ ×[0,1]. To see that such a plane cannot exist, we first note that it cannot contain any of the four squares of Γ ×[0,1] with xor ysides, as each side of this type occurs exactly once in Γ ×[0,1], so that it is not possible to extend Pacross such an edge. It follows then that no square with an rside can be contained in P, as such a square must be adjacent across its redge to a square with an xor yedge. Thus Pconsists entirely of squares from the “middle” section of Γ ×[0,1]. Fix a point uin the interior of such a square, and consider a lift in Pof the “vertical” segment u×[0,1] ⊂Γ×[0,1]. Because Pconsists entirely of squares from Γ ×[0,1], this segment lifts to a bi-infinite line in P. On the other hand, moving in the positive direction along this line, one successively crosses successive biedges before finally intersecting baq, at which point the line cannot be extended in Γ ×[0,1]. It follows that no such plane Pexists, so π(C(X)) is hyperbolic.  Remark 4.1. We mentioned in the introductory paragraph to this section that the ambient free-by-cyclic group in this example is of the form F65 ⋊Z. Here is one way to see this. From Proposition 1.5 we know that the rank of the free kernel is equal to the rank of the preimage of the basepoint of the target circle. This preimage is a graph with 12 vertices: v1,v2, two points dividing the edges r1, r2, s2into thirds, and four points dividing s1