The Closure of a Random Braid Is a Hyperbolic Link

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 142, Number 2, February 2014, Pages 695–701 S 0002-9939(2013)11775-2 Article electronically published on October 31, 2013

THE CLOSURE OF A RANDOM BRAID IS A HYPERBOLIC LINK

JIMING MA

(Communicated by Daniel Ruberman)

Abstract. Let μ be a probability distribution on the braid group Bn with n ≥ 3 strands. We observe that for a random walk ωn,k of length k on Bn, 3 the probability that the closure ωn,k is a hyperbolic link in S converges to 1ask tends to inﬁnity. Moreover, under a mild assumption on μ,weprove the probability that the closure ωn,k is a hyperbolic knot which has no non- trivial exceptional surgeries is larger than zero for k large enough. The proofs combine several recent deep results.

1. Introduction It is well-known (for example, see [1]) that there are three types of knots in S3: torus knots, satellite knots and hyperbolic knots. Similarly there is a rough classification of links in S3, which is a little more complicated; i.e., we should also consider split links. According to the Geometrization Theorem for Haken Manifolds of Thurston ([9,21]), most links in S3 should be hyperbolic. To describe this statement more precisely, we need the notion of random links in S3. There are several models of random links: for example, via random walks on braid groups or via random bridge presentations of links, and there is a more combinatorial model given by a uniformly distributed collection of unit length sticks stuck end to end [4]. The question is whether random links via these models are hyperbolic. Unfortunately, a random knot via the third model is not hyperbolic; actually, with a high probability the knot is a satellite knot [8]. Thus, the hyperbolicity of a random link is a non-trivial question. In this note, we address this question via the random braid model: Theorem 1.1. Let μ be a non-elementary probability distribution with finite first moment on the braid group Bn with n ≥ 3 strands. Then for a random walk ωn,k 3 of length k on Bn, the probability that the closure ωn,k is a hyperbolic link in S converges to 1 as k →∞. This result should be compared to [15], where Malyutin uses quasi-morphisms and some results tracing back to works of Bestvina-Fujiwara, J. Birman, I. Dyn- nikov, Malyutin-Netsvetaev and W. Menascotoshowthatforanon-degenerate random walk on Bn, the subset of Bn consisting of braids whose closures are non- prime links is transient. In particular, for a non-degenerate random walk on Bn,

Received by the editors September 18, 2011 and, in revised form, February 10, 2012 and March 25, 2012. 2010 Mathematics Subject Classiﬁcation. Primary 57M25, 57M50, 20F36. Key words and phrases. Hyperbolic link, braid group, random walk, Dehn surgery. The author was supported in part by NSFC 10901038 and NSFC 11371094.

c 2013 American Mathematical Society 695

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 696 JIMING MA

the asymptotic probability that the closure of it is a hyperbolic link, a torus link or a link with essential torus but without essential meridian annulus and sphere in its complement is 1. The author does not know whether Malyutin’s method can be sharpened to show that the probability of the hyperbolicity of a random link is 1. We remark that the (more difficult) question of whether a random closed 3- manifold is hyperbolic was solved affirmatively by J. Maher [13] and Lustig-Moriah [11, 12] with the help of the Geometrization Theorem for 3-manifolds via different models of random Heegaard splittings. If M is a finite volume hyperbolic 3-manifold with exactly one torus cusp, then by the Hyperbolic Dehn Surgery Theorem [18, 21], all but finitely many Dehn fillings of M result in closed hyperbolic 3-manifolds, and a slope with which the Dehn filling is performed such that the resultant is a non-hyperbolic 3-manifold is called 3 an exceptional slope.LetL = {K1,K2,...,Kn} be a hyperbolic link in S . Recall that there is a canonical way to parameterize the space of (multiple) Dehn fillings 3 of S − intN(L): we fix a homology basis (mi,li) for each boundary component 3 of S − intN(L), where mi bounds a disk in N(Ki)andli is homologically zero 3 in S − intN(Ki). Thus, if a slope represents (pmi + qli) in the first homology of ∂N(Ki), we have an element pi/qi of Q ∪∞ uniquely defined by the slope, and conversely. We call (∞, ∞,...,∞)thetrivial slope for S3 −intN(L)(orL for short), and so any n-tuple α =(a1,a2,...,an) =( ∞, ∞,...,∞)isnon-trivial. Moreover, if none of ai is ∞,thenwesaya is strictly non-trivial. Since the Dehn filling along (∞, ∞,...,∞)resultsinS3, there is always an exceptional slope for L. Purcell [19] and Futer-Purcell [5] formulate combinatorial conditions on the diagram of a link L in S3 which ensure that all strictly non-trivial slopes are non-exceptional slopes. We show that links in S3 with this property are overwhelming:

Proposition 1.2. Let μ be a non-elementary probability distribution on the braid group Bn with n ≥ 3 strands. Then for a random walk ωn,k of length k on Bn,the 3 probability that the closure ωn,k is a hyperbolic link in S such that every strictly non-trivial slope is non-exceptional converges to 1 as k →∞.

Since it is possible that a Dehn filling along a slope α which is non-trivial but also not strictly non-trivial results in a hyperbolic 3-manifold when L is a link with at least two components. For example, when L = {K1,K2}, the Dehn filling along a slope (∞,p/q) =( ∞, ∞) may result in a hyperbolic 3-manifold if K2 is hyperbolic. We consider the probability that for a random link in S3, all but finitely many Dehn fillings in the (multiple) Dehn filling space result in hyperbolic 3-manifolds. We want to know whether this probability is one. (Of course, there are links which admit infinitely many non-hyperbolic Dehn fillings; for example, see [16].) For instance, if L = {K1,K2} is a hyperbolic link such that the meridian in each cusp has small normalized length (see Section 2), and we also assume Ki is a hyperbolic knot with the same condition on normalized length for i =1, 2, then it is easy to see all non-trivial slopes of L are non-exceptional, and then there is exactly one exceptional slope for L. We expect that links with only finitely many exceptional slopes are prevalent in S3. We cannot prove the probability is one. But under a mild condition on the random walk, we can show that the probability that the closure of a random braid is a

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE CLOSURE OF A RANDOM BRAID IS A HYPERBOLIC LINK 697

knot without non-trivial exceptional slopes is positive. In particular, the probability that the closure of a random braid is a link with only finitely many exceptional slopes is positive. There is a natural map π from Bn to Sn, the permutation group on n letters Ω={1, 2,...,n}:letσ1,σ2,...,σn−1 be the canonical generators of Bn—that is, when viewing Bn as the mapping class group of a disk with n punctures, σi is the (right) half-twist which permutes the i-th and (i + 1)-th punctures—and let si be the simple transposition on the letters i and i+1 for1 ≤ i ≤ n−1; then π(σi)=si. Thus, a probability distribution μ on Bn induces a probability distribution on Sn (we denote it also by μ for simplicity), and then a random walk on Bn induces a random walk on Sn. The number of components of our link ωn,k is just the number of orbits of π(ωn,k)onΩ={1, 2,...,n}, i.e., the number of a presentation of π(ωn,k) as a product of disjoint cycles. The probability that ωn,k is a knot is exactly the probability that π(ωn,k)isann-cycle. It is unknown to the author whether a generic random walk on Sn is uniformly distributed, i.e., whether or not P(wk = s)convergesto1/n!ask diverges to infinity for any s ∈ Sn. This question seems non-trivial. For example, consider the canonical probability distribution on Sn,whereμ(si)=1/(n − 1). In this case, only the random walk ω2k can arrive at the identity; i.e., we must take care of parity issues. I. Rivin [20] considers this question for random walks on finite groups. But unfortunately, the canonical random walk on Sn does not satisfy the second term of Condition 11.1 of [20]; i.e., the condition on 1-dimensional unitary representations. The author also does not know of any general condition which ensures that the probability that a random walk on Sn arrives at an n-cycle is positive, which in turn ensures the positive probability of knots in the closures of random braids. We consider a mild condition on the probability distribution μ on Bn,whichis motivated by [3], such that its projection to Sn satisfies 2 (1.1) μ(e =1/n)andμ(si,j =2/n ),

where si,j is the transposition on the letters i and j with i = j. Theorem 1.3. Let μ be a non-elementary probability distribution on the braid group Bn with n ≥ 10 strands which satisﬁes assumption (1.1). Then for a random walk ωn,k of length k on Bn, the probability that the closure ωn,k is a hyperbolic knot in S3 without non-trivial exceptional slopes is larger than zero when k is large enough.

2. Proofs Let G be the mapping class group (or a subgroup of the mapping class group which preserves some preferred punctures) of a surface F with negative Euler char- acteristic. A probability distribution μ on G is non-elementary if the semigroup in G generated by the support of μ contains two independent pseudo-Anosov ele- ments. We don’t give the precise definition of a finite first moment here; see [23]. But we remark that if the support of μ is finite, then it has a finite first moment, and moreover if the support of μ generates a finite index subgroup of G,thenμ is non-elementary. The probability distribution μ generates a random walk (Markov chain) on G, with transition probability P(x, y)=μ(x−1y), which we assume starts at the identity element of G.Wedenotebyωk the random variable corresponding to the k-step

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 698 JIMING MA

∗k random walk. Thus, P(ωk ∈ X) is given by the k-fold convolution μ of μ for a subset X in G. In particular, if the support of μ is ﬁnite, then P(ωk = x)= Σμ(x1)μ(x2) ...μ(xk), where the summand ranges over all tuples (x1,x2,...,xk) with x1x2 ...xk = x.

Proof of Theorem 1.1. Our proof of Theorem 1.1 is based on the recent results of [6], [14] and on the behavior of Dehn fillings along long slopes [2, 7, 10]. Here we use the notion of normalized length [7], which is more convenient for our purpose. Recall that a cusp in a finite volume hyperbolic 3-manifold M is topologically T × [0, ∞), where T a torus, and any T ×{x} admits a Euclidean structure τx.The Euclidean structure on T ×{y} is a re-scaling of the Euclidean structure on T ×{x} for different x and y in [0, ∞). For a slope λ in T ,thenormalized length of λ ([7]) is ×{ } (2.1) lN (λ)=Lengthτx (λ)/ Area(T x );

then lN (λ) is independent of the choice of x ∈ [0, ∞). There are two complexes associated to a surface F (for our purpose, we assume F is a sphere with at least 4 punctures), the curve complex C (F )andthearc complex A (F, p) with some preferred cusps p.Thecurve complex C (F )isthe graph whose vertices are isotopy classes of essential curves in F .IfF is not a 4-punctured sphere, then two vertices are connected by an edge of length 1 if and only if they have disjoint representatives, while in the 4-punctured sphere case two vertices are connected by an edge of length 1 if and only if they have representatives which intersect in two points. The arc complex A (F, p) with some preferred cusps p is the graph whose vertices are isotopy classes of essential arcs in F with at least one endpoint in the preferred punctures p, and two vertices are connected by an edge of length 1 if and only if there are representations of them which are disjoint. For an arc x in F ,letb be those punctures in F corresponding to the ends of x. Then the boundary of a neighborhood of x ∪ b,say∂N(x ∪ b), is one or two curves in F , at least one of which is essential in F . Denote the essential components of ∂N(x ∪ b)bycx,socx has one or two components. If x and y are two arcs which are disjoint in F ,thencx intersects cy in at most two points. Thus, any component ∈ ∈ C cx cx and any component cy cy have distance at most 2 in (F ) by elementary surgeries on curves. Recall that for a map f in the mapping class group of F ,thestable translation length of f on A (F, p)is

m (2.2) lA (F,p)(f) = lim dA (F,p)(x, f (x))/m, m→∞

for any arc x in A (0)(F, p), which is well-defined independent of the choice of x. A similar definition holds for the curve complex. The stable translation length is larger than zero if and only if f is a pseudo-Anosov map. The main result of [14] asserts that the stable translation length of a random walk on the mapping class group of F on C (F ) increases linearly: there are two positive constants α and β such that the probability P(βk ≤ lC (F )(ωk) ≤ αk) converges to one as k diverges to infinity. (Maher works on a relative metric of the mapping class group, which is quasi-isometric to C (F ), so we just have two

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE CLOSURE OF A RANDOM BRAID IS A HYPERBOLIC LINK 699

constants α and β, but we don’t know whether they can be chosen asymptotically equal.) In particular, a random walk ωk is a pseudo-Anosov map in the mapping class group with asymptotic probability one. m m For any arc x,wehavedA (F,p)(x, f (x)) ≥ dC (F )(cx,f (cx))/2 by the previ- ous paragraphs, so lA (F,p)(ωk) ≥ lC (F )(ωk). In particular, for any N ∈ N,the probability P(lA (F,p)(ωk) ≥ N)convergesto1ask →∞. Now let wn,k be a random walk of length k on Bn,andwn,k be the closure of 3 wn,k in S . We view our link wn,k as a Dehn filling on a fibered 3-manifold M with fiber an (n + 1)-punctured sphere F :abraidwn,k corresponds to a mapping class in an n-punctured disk D which fixes ∂D pointwisely, and we view the n- punctured disk as an (n + 1)-punctured sphere F with a preferred puncture p;the preferred puncture p is just the boundary of the disk. The fibered 3-manifold M is D × [0, 1]/ ∼, where the equivalence relation ∼ is (z,0) = (wn,k(z), 1). M is 3 hyperbolic if and only if wn,k is pseudo-Anosov ([17, 22]). S − wn,k is the Dehn filling on M along the slope S1 = {z}×[0, 1]/ ∼. Now from Theorem 1.5 of [6] we have that for any slope λ in ∂D × [0, 1]/ ∼ except the longitude ∂D ×{t} (the longitude is defined from the viewpoint of fibered 3-manifolds, and it is the meridian from the viewpoint of links in S3), the normalized length of λ is bounded below by 4 lA (wn,k)/(536χ(F ) ) lA (wn,k) (2.3) (F,p) = (F,p) . 2 5 9lA (F,p)(wn,k)χ(F ) 1608χ(F )

In particular, the normalized length of {z}×[0, 1]/ ∼ is suﬃciently large with 3 probability one as k →∞. Then by Theorem 5.11 of [7], S − wn,k is hyperbolic (we actually use a variation of Theorem 5.11 of [7], i.e., when there is a Dehn ﬁlling along one cusp and the remaining cusps remain open).

Proof of Proposition 1.2. Now the proof of this proposition runs along the same lines as Theorem 1.1. We consider another manifold M = D ×[0, 1]/ ∼,wherethe ∼ n! equivalence relation is (z,0) = (wn,k(z), 1). M is a cover of M and has exactly (n+1) cusps. The lower bound on the normalized length of a non-longitude slope on every cusp of M gives a lower bound on the normalized length of a non-longitude slope on every cusp of M, up to the multiplicative factor n!. 3 A Dehn filling on S − wn,k is just a special Dehn filling on the fibered manifold M, where one factor of the Dehn filling slope is fixed. Theorem 1.5 of [6] gives a lower bound on the normalized length for each factor of a strictly non-trivial slope 3 of S − wn,k. Then the probability that the normalized length of each factor of a strictly non-trivial slope on M is sufficiently large converges to 1 when k is large. Then, again by the theorem of Hodgson-Kerckhoff, we are done.

Proof of Theorem 1.3. Under assumption (1.1), we can use Theorem 1 of [3], which asserts that the distance between the k-fold convolution μ∗k and the uniform distribution decays roughly exponentially with k when n ≥ 10. In particular, μ∗k nearly uniformly distributes when k is large. There are (n − 1)! n-cycles in Sn,so the probability of a random walk ωn,k on Bn results in a knot that converges to (n − 1)!/n!=1/n as k diverges to inﬁnity.

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 700 JIMING MA

Acknowledgements The author would like to thank the referee and Daniel Ruberman for comments on an early draft.

References

[1] Colin C. Adams, The knot book. An elementary introduction to the mathematical theory of knots, W. H. Freeman and Company, New York, 1994. MR1266837 (94m:57007) [2] Ian Agol, Bounds on exceptional Dehn filling,Geom.Topol.4 (2000), 431–449, DOI 10.2140/gt.2000.4.431. MR1799796 (2001j:57019) [3] Persi Diaconis and Mehrdad Shahshahani, Generating a random permutation with random transpositions,Z.Wahrsch.Verw.Gebiete57 (1981), no. 2, 159–179, DOI 10.1007/BF00535487. MR626813 (82h:60024) [4] Yuanan Diao, Nicholas Pippenger, and De Witt Sumners, On random knots, Random knotting and linking (Vancouver, BC, 1993), Ser. Knots Everything, vol. 7, World Sci. Publ., River Edge, NJ, 1994, pp. 187–197, DOI 10.1142/9789812796172 0013. MR1474271 (98e:57010) [5] David Futer and Jessica S. Purcell, Links with no exceptional surgeries, Comment. Math. Helv. 82 (2007), no. 3, 629–664, DOI 10.4171/CMH/105. MR2314056 (2008k:57008) [6] David Futer and Saul Schleimer, Cusp geometry of fibered 3-manifolds,arXiv: math.GT/1108.5748, to appear in Amer. J. Math. [7] Craig D. Hodgson and Steven P. Kerckhoff, The shape of hyperbolic Dehn surgery space, Geom. Topol. 12 (2008), no. 2, 1033–1090, DOI 10.2140/gt.2008.12.1033. MR2403805 (2010b:57021) [8] Douglas Jungreis, Gaussian random polygons are globally knotted,J.KnotTheoryRamifica- tions 3 (1994), no. 4, 455–464, DOI 10.1142/S0218216594000332. MR1304395 (95i:57008) [9] Michael Kapovich, Hyperbolic manifolds and discrete groups, Progress in Mathematics, vol. 183, Birkhäuser Boston Inc., Boston, MA, 2001. MR1792613 (2002m:57018) [10] Marc Lackenby, Word hyperbolic Dehn surgery, Invent. Math. 140 (2000), no. 2, 243–282, DOI 10.1007/s002220000047. MR1756996 (2001m:57003) [11] Martin Lustig and Yoav Moriah, Horizontal Dehn surgery and genericity in the curve complex, J. Topol. 3 (2010), no. 3, 691–712, DOI 10.1112/jtopol/jtq022. MR2684517 (2011k:57028) [12] Martin Lustig and Yoav Moriah, Are large distance Heegaard splittings generic?,withan appendix by Vaibhav Gadre, J. Reine Angew. Math. 670 (2012), 93–119. MR2982693 [13] Joseph Maher, Random Heegaard splittings, J. Topol. 3 (2010), no. 4, 997–1025, DOI 10.1112/jtopol/jtq031. MR2746344 (2012d:37087) [14] Joseph Maher, Linear progress in the complex of curves, Trans. Amer. Math. Soc. 362 (2010), no. 6, 2963–2991, DOI 10.1090/S0002-9947-10-04903-2. MR2592943 (2011e:20063) [15] A. V. Malyutin, Quasimorphisms, random walks, and transient subsets in countable groups, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 390 (2011), Teoriya Predstavlenii, Dinamicheskie Sistemy, Kombinatornye Metody. XX, 210–236, 309–310 (Eng- lish, with English and Russian summaries); English transl., J. Math. Sci. (N. Y.) 181 (2012), no. 6, 871–885, DOI 10.1007/s10958-012-0721-7. MR2870236 (2012k:60140) [16] Bruno Martelli and Carlo Petronio, Dehn filling of the “magic” 3-manifold, Comm. Anal. Geom. 14 (2006), no. 5, 969–1026. MR2287152 (2007k:57042) [17] Jean-Pierre Otal, Le théorème d’hyperbolisation pour les variétés fibrées de dimension 3, Astérisque (235), 1996. Translated into English as Jean-Pierre Otal, Leslie D. Kay, The hy- perbolization theorem for fibered 3-manifolds, SMF/AMS Texts and Monographs, American Mathematical Society, 2001. MR1402300 (97e:57013) [18] Carlo Petronio and Joan Porti, Negatively oriented ideal triangulations and a proof of Thurston’s hyperbolic Dehn filling theorem,Expo.Math.18 (2000), no. 1, 1–35. MR1751141 (2001c:57017) [19] Jessica S. Purcell, Cusp shapes of hyperbolic link complements and Dehn filling,ProQuest LLC, Ann Arbor, MI, 2004. Thesis (Ph.D.)–Stanford University. MR2705870 [20] Igor Rivin, Walks on groups, counting reducible matrices, polynomials, and surface and free group automorphisms, Duke Math. J. 142 (2008), no. 2, 353–379, DOI 10.1215/00127094- 2008-009. MR2401624 (2009m:20077)

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE CLOSURE OF A RANDOM BRAID IS A HYPERBOLIC LINK 701

[21] William P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 2, 417–431, DOI 10.1090/S0273-0979-1988-15685-6. MR956596 (89k:57023) [22] William P. Thurston, Hyperbolic structures on 3-manifolds, II: surface groups and 3- manifolds which fiber over the circle, arXiv: math.GT/9801045, 1998. [23] Wolfgang Woess, Random walks on infinite graphs and groups, Cambridge Tracts in Mathe- matics, vol. 138, Cambridge University Press, Cambridge, 2000. MR1743100 (2001k:60006)

School of Mathematical Science, Fudan University, Shanghai, People’s Republic of China, 200433 E-mail address: [email protected]

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use