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SUPER-EXPONENTIAL DISTORTION OF SUBGROUPS OF CAT(−1) GROUPS 7 JOSH BARNARD,NOEL BRADY,ANDPALLAVIDANI 0 0 2 Abstract. We construct 2-dimensional CAT(-1) groups which contain free subgroups n with arbitrary iterated exponential distortion, and with distortion higher than any iter- a ated exponential. J 1 3 ] 1. Introduction R G The purpose of this note is to produce explicit examples of CAT(-1) groups containing free subgroups with arbitrary iterated exponential distortion, and distortion higher than . h any iterated exponential. The construction parallels that of Mahan Mitra in [Mi] but our t a groups are the fundamental groups of locally CAT(−1) 2-complexes. The building blocks m used in [Mi] are hyperbolic F ⋊ F groups, which are not known to be CAT(0). Our 3 3 [ building blocks are graphs of groups where the vertex and edge groups are all free groups 1 ofequalrankandtheunderlyinggraphisabouquetofafinitenumberofcircles. Weusethe v combinatorial and geometric techniques from Dani Wise’s version of the Rips construction 0 [Wi] to ensure that our building blocks glue together in a locally CAT(−1) fashion. 3 9 One of our motivations for producing these examples was that it was not immediate 1 from the description that the examples in [Mi] had the appropriate iterated exponential 0 distortions. Mitra’s examples are graphs of groups with underlying graph a segment of 7 0 length n, where the vertex groups are hyperbolic F3⋊F3 groups, and each edge identifies / the kernel F in one vertex group with the second F factor in the adjacent vertex group. h 3 3 t While it is easy to see that the nth power of a hyperbolic automorphism of a free group a will send a generator of the free group into a word which grows exponentially in n, it m appears to be harder to see (without using bounded cancellation properties of carefully : v chosen automorphisms) that a word of length n in three hyperbolic automorphisms (and i X theirinverses)willsendageneratorofthefreegrouptoawordwhichgrowsexponentiallyin n. Incontrast, themonomorphismsinthemultipleHNNextensions inourconstruction are r a all defined using positive words. This makes it easy to see that the exponential distortions compose as required. Also, the example in [Mi] with distortion higher than any iterated exponential is of theform (F ⋊F )⋊Z,with thegenerator of Zconjugating thegenerators 3 3 ofthefirstF to“sufficientlyrandom”wordsinthegeneratorsofthesecondF . Incontrast, 3 3 our group can be described explicitly, without recourse to random words, allowing for an explicit check that our group is CAT(−1). Noel Brady acknowledges support from NSFgrant no. DMS-0505707. 1 2 JOSHBARNARD,NOELBRADY,ANDPALLAVIDANI Recall that if H ⊂ G is a pair of finitely generated groups with word metrics d and d H G respectively, the distortion of H in G is given by δG(n)= max{d (1,h) |h ∈ H with d (1,h) ≤ n}. H H G Up to Lipschitz equivalence, this function is independent of the choice of word metrics. Background on CAT(−1) spaces and the large link condition may be found in [BH]. 2. The building blocks For each positive integer n we define a building block group G = ha ,...,a ,t ,...,t | t a t−1 = W ; 1≤ i ≤ n,1 ≤ j ≤ mi, n 1 m 1 n i j i ij where m = 14n and {W } is a collection of positive words of length 14 in a ,...,a , such ij 1 n thateach two-letter word a a appearsat mostonceamong theW ’s. Oneway of ensuring k l ij this is to choose these words to be consecutive subwords of the following word, defined by Dani Wise in [Wi]. Definition 2.1 (Wise’s long word with no two-letter repetitions). Given the set of letters {a ,...,a }, define 1 m Σ(a ,...,a ) = (a a a a a ···a a )(a a a a a ···a a )······(a a a )a . 1 m 1 1 2 1 3 1 m 2 2 3 2 4 2 m m−1 m−1 m m It is easy to see that Σ(a ,...,a ) is a positive word of length m2, such that each 1 m two-letter word a a appears as a subword in at most one place. Following Dani Wise, we k l simply chop Σ(a ,...,a ) into subwords of length 14. In order to obtain all mn relator 1 m words of G from Σ(a ,...,a ) we must have m2 ≥ 14mn, which explains our choice of n 1 m m above. Proposition 2.2. The presentation 2-complex X for G can be given a locally CAT(−1) n n structure. Furthermore, (1) The a ’s generate a free subgroup F(a ) whose distortion in G is exponential. j j n (2) The t ’s generate a free subgroup F(t ) of G that is highly convex in the following i i n sense: Let v be the vertex of X . Then n d (tǫ1,tǫ2)≥ 2π, where ǫ ,ǫ ∈ {+,−} and if i= j then ǫ 6= ǫ . Lk(v,Xn) i j 1 2 1 2 t a t i j i W ij Figure 1. A 2-cell of X decomposed into right-angled pentagons. n SUPER-EXPONENTIAL DISTORTION OF SUBGROUPS OF CAT(−1) GROUPS 3 Proof. Each disk in X is given a piecewise hyperbolic structure by expressing it as a n concatenation of right-angled hyperbolic pentagons, as shown in Figure 1. The fact that X satisfies the large link condition is a consequence of the condition that the W ’s are n ij positive words with no two-letter repetitions. The details are in [Wi]. Thus X is locally n CAT(−1) and G is hyperbolic. n To prove (1), first note that the group G can be viewed as the fundamental group of a n graph of groups. The underlying graph is a bouquet of n circles, and the edge and vertex groups are all equal to F(a ), the free group on a ,...,a . The two maps associated with j 1 m the ith edge are id and φ :F(a )→ F(a ), defined by φ (a )= W for 1≤ j ≤ m. To F(aj) i j j i j ij see that φ is injective, note that φ induces a map on a subdivided bouquet of m circles. i i By Stallings’ algorithm this factors through a sequence of folds followed by an immersion. The no two-letter repetitions condition restricts the amount of folding that can occur and ensures that no non-trivial loops are killed by the sequence of folds. Thus the folding maps induce isomorphisms at the level of π and the final immersion induces an injection. The 1 theory of graphs of groups now implies that the subgroup of G generated by a ,...,a n 1 m is F(a ). j The distortion of F(a ) in G is at least exponential because, for example, the element j n tka t−k of F(a ), when expressed in terms of the a ’s, is a positive (hence reduced) word 1 1 1 j j of length 14k. To see that the distortion is at most exponential consider a word w(a ,t ) j i which represents an element of F(a ) and has length k in G . It can be reduced to a word j n in the a ’s by successively cancelling at most k/2 innermost t ···t−1 pairs. Each such j i i cancellation multiplies the word length by at most a factor of 14. So the length of w(a ,t ) j i in F(a ) is at most 14k/2k. j To prove (2), note that any path in the link of v connecting t± and a± within a disk as i j in Figure 1 has combinatorial length 2. So any path in the link connecting tǫ1 and tǫ2 has i j combinatorial length at least 4. Since we are using right-angled pentagons, combinatorial length 4 corresponds to a spherical metric length of 4(π/2) or 2π. Thus ht ,...,t i is 1 n highly convex. As a consequence, we have that the map from the bouquet of circles with edges labelled t ,...,t into X is a local isometric embedding. This implies that it is 1 n n π -injective, i.e. t ,...,t generate the free group F(t ). To see this algebraically, note 1 1 n i that the homomorphism ψ :G → F(t ) defined byψ(t ) = t and ψ(a ) = 1 is a retraction n i i i j of G onto F(t ). (cid:3) n i 3. Iterated exponential distortion Inthissectionweseehowtogetarbitraryiteratedexponentialdistortions. Theideaisto amalgamate a chain of building block groups together, identifying the distorted free group in one with the highly convex free subgroup in the next. This identification of distorted withhighlyconvexcanbemadeinanon-positivelycurvedway,andthedistortionfunctions compose as expected. Here are the details. Theorem 3.1. For any integer l > 0, there exists a 2-dimensional CAT(−1) group H l with a free subgroup F such that δHl(x) ≃ expl(x). F 4 JOSHBARNARD,NOELBRADY,ANDPALLAVIDANI We will actually show that δHl(x) ≃ fl(x), where f(x) = 14x. F Proof. The group H is defined using the building blocks from Proposition 2.2 as l Hl = G1∗F1 G14 ∗F2 G142 ∗······∗Fl−1 G14l−1, where F , for 1 ≤ k ≤ l − 1, is a free group of rank 14k which is identified with the k exponentially distorted free subgroup of G14k−1 and the highly convex free subgroup of G14k. Let a(jk), with 1 ≤ j ≤ 14k, denote the generators of Fk, and let t denote the stable (1) (k+1) (k) letter of G1, so that G1 = haj ,ti and G14k = haj ,aj i. We shall use this notation in the upper and lower bound arguments below. Let Y denote the presentation complex of H . Then Y = X , which is locally CAT(−1) l l 1 1 by Proposition 2.2. Further, there are inclusions Y ⊂ Y ⊂ Y ···, so that the large link 1 2 3 condition can be checked inductively. The space Yk+1 is obtained by gluing X14k to Yk along a rose Rk with 14k petals, and Rk is highly convex in X14k. Itfollows that the link of the base vertex of Y is obtained by gluing together the links of base vertices in Y and k+1 k X14k, along a set of 2(14k) points which is 2π-separated in the latter link. By induction, the link of the base vertex in Y is large, and hence the union is large. k (l) The group with the desired distortion in H is F , the free group generated by a , with l l j 1 ≤ j ≤ 14l, which is exponentially distorted in G14l−1. We prove the lower bound as follows. Given a positive integer n consider the sequence given by w = tna(1)t−n ∈ F and 1 1 1 w = w a(k)w−1 ∈ F , and set w = w . Given g ∈ H let ℓ (g) denote the distance k (k−1) 1 (k−1) k l l Hl from1tog inH . Observethatℓ (w ) ≤ 2n+1, andinductively thatℓ (w) ≤ 2ln+2l−1 l Hl 1 Hl (a linear function of n). On the other hand, each w can be expressed as a positive word k in the generators of F by using the W ’s from the definition of the building blocks. Since k ij there is no cancellation among positive words, we obtain that |w | = 14n and inductively 1 F1 that |w| = fl(n). This gives the lower bound fl(x)(cid:22) δHl(x). Fl F To prove the upper bound it is more convenient to do the induction in the opposite direction. Proposition 2.2 provides the base case. Let w be an element whose length in H l is at most n. Then by successively cancelling at most n/2 innermost t···t−1 pairs, w can be represented by a word in G14 ∗F2 G142 ∗···G14l−1 of length at most 14n/2n. At each stage of the previous cancellation procedure, we may assume (by replacement if necessary) that the subword enclosed by an innermost t...t−1 involves only the a(1). This is because i (1) the free group on the ai is a retract of G14∗F2G142∗···G14l−1. The upper bound follows by induction. (cid:3) 4. Distortion higher than any iterated exponential In this section we produce a 2-dimensional CAT(-1) group containing a free subgroup with distortion more than any iterated exponential. The idea is to take a suitable building block group from section 2 with base group free on a and stable letters t , and to form i j SUPER-EXPONENTIAL DISTORTION OF SUBGROUPS OF CAT(−1) GROUPS 5 a new HNN extension with stable letter s which sends the free group on the a into a i subgroup of the free group on the t . j Theorem 4.1. There exists a 2-dimensional CAT(−1) group G with a subgroup H, such that δG(x) is a function that is bigger than any iterated exponential. H t a t i j i W (a) ij s a s k W (t) k Figure 2. Relator 2-cells of the group G decomposed into right-angled pentagons. Proof. Define G = (cid:10)a1,...,am,t1,...,tn,s |tiajt−i 1 = Wij ; saks−1 = Wk(cid:11), where {W } (resp. {W }) consists of mn (resp. m) positive words of length 14 in the ij k letters a (resp. t ), with no 2-letter repetitions. Thus we may choose the W ’s and W ’s i j ij k to be disjoint subwords of Σ(a ,...,a ) and Σ(t ,...,t ) (see Definition 2.1) respectively. 1 m 1 n This gives the following two conditions: 14mn ≤ m2 and 14m ≤ n2. So, for example, n = 142 and m = 143 is a possible choice. Each disk in the presentation complex is given a piecewise hyperbolic structure by con- catenating a sequence of right-angled pentagons as shown in Figure 2. Observe that loops in the link of the unique vertex which involve s± have combinatorial length at least 4. Similarly, any loop involving an a± and a t±, for some i and j, has length at least 4. Thus j i to show that the complex is locally CAT(−1), it is enough to consider loops involving only a±’s or only t±’s. The fact that such loops are large is a consequence of the condition i j that the W ’s and W ’s are positive words with no two-letter repetitions. The details are ij k exactly as in [Wi]. 6 JOSHBARNARD,NOELBRADY,ANDPALLAVIDANI w w 1 1 w2 w2 w1 w1 w 3 Figure 3. The word w (in this case k = 3). k Let H be the subgroup of G generated by a ,...,a . Then δG(x) is higher than any 1 m H iterated exponential. To see this, consider the sequence of words w ∈ H given by w = k 1 t a t−1 and w = (sw s−1)a (sw−1 s−1) for k > 1. The word w is shown in Figure 3. 1 1 1 k k−1 1 k−1 3 The label of each single arrow edge is a , the label of each solid arrow edge is t , and the 1 1 edges along the strips are all labeled by s and are oriented from w toward w . Let j j−1 ℓ and ℓ denote geodesic lengths in G and H respectively. Note that ℓ (w ) = 3 and G H G 1 ℓ (w ) ≤ 2ℓ (w )+5fork > 1. So,forexample,ℓ (w ) ≤ 4k istrue. Ontheotherhand, G k G k−1 G k ℓH(w1) = 14 and ℓH(wk) = 1414ℓH(wk−1) > 14ℓH(wk−1). So by induction ℓH(wk) ≥ fk(1). (Recall that f(x) = 14x.) This shows that δG(x) ≥ f⌊log4x⌋(1), which is a function that H grows faster than any iterated exponential. (cid:3) References [BH] M. R. Bridson and A. Haefliger, Metric spaces of non-positive curvature. Grundlehren der Mathe- matischen Wissenschaften, 319. Springer-Verlag, Berlin, 1999. [Mi] M. Mitra, Cannon-Thurston maps for trees of hyperbolic metric spaces, J. Diff. Geom. 48 (1998), no. 1, 135–164. [Wi] D.Wise, Incoherent negatively curved groups, Proc. Amer. Math. Soc. 126 (1998), no. 4, 957–964. Dept. of Mathematics, University of Oklahoma, Norman, OK 73019 E-mail address: jbarnard, nbrady, pdani @math.ou.edu