A CENTRAL LIMIT THEOREM FOR RANDOM CLOSED GEODESICS: PROOF OF THE CHAS LI MASKIT CONJECTURE

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1 A CENTRAL LIMIT THEOREM FOR RANDOM CLOSED GEODESICS: PROOF OF THE CHAS LI MASKIT CONJECTURE ILYA GEKHTMAN, SAMUEL J. TAYLOR, AND GIULIO TIOZZO Abstract. We prove a cetral limit theorem for the legth of closed geodesics i a hyperbolic pair of pats, thus settlig a cojecture of Chas Li Maskit. Let Σ be a pair of pats with a fixed hyperbolic metric such that each compoet of Σ is a geodesic. Its fudametal group G is a free group o two geerators. Let S be a stadard geeratig set of G i.e., a free basis, ad let g be the word legth of g with respect to S. For each g G, let [g] deote its cojugacy class, ad for ay cojugacy class γ = [g] defie its cojugacy legth γ = g := mi [g]=γ g to be the miimum word legth over all elemets represetig γ. Ay cojugacy class γ is represeted by a closed geodesic i the pair of pats, ad let τγ deote the legth of such geodesic i the hyperbolic metric. Let µ deote the uiform distributio o the set F of cojugacy classes of legth. The goal of this ote is to prove the followig cetral limit theorem: Theorem 1. There exist costats L > 0, σ > 0 such that for ay a, b R with a < b we have τγ L µ γ : σ [a, b] 1 b e x2 2 dx 2π a as. This is the mai cojecture of Chas Li Maskit i [4], motivated by experimetal evidece. The proof combies tools of Chas Lalley [3], Calegari [1], ad the authors [5]. The same techiques ca be used to exted Theorem 1 to all orietable compact hyperbolic surfaces, but we will ot discuss this at the momet. We ote that P. Park has recetly writte up a related result where the uiform distributio o cojugacy classes is replaced by the th step distributio of a simple radom walk o G [7]. Prelimiary results. By costructio we have that Σ is the covex core of H 2 /G, where G acts by isometries as a Schottky group o the hyperbolic plae H 2. Fix a base poit x H 2. The for γ = [g], τγ = τg equals the stable traslatio legth of g o H 2. Hece, oe has the formula see e.g. [6, Propositio 5.8] 1 τg = dgx, x 2gx, g 1 x x + Oδ Date: August 25, Gekhtma is partially supported by NSF grat DMS Taylor is partially supported by NSF grat DMS Tiozzo is partially supported by NSERC ad the Alfred P. Sloa Foudatio. 1

2 2 I. GEKHTMAN, S.J. TAYLOR, AND G. TIOZZO where x, y z deotes the Gromov product, ad A = B + Oδ meas that there is a costat C, which depeds oly o the hyperbolicity costat δ of H 2, such that A B C. Let S be the set of reduced words of legth, ad let J be the set of cyclically reduced words of legth. I particular, these are the same as the set of reduced words of legth whose first letter is ot the iverse of the last letter. Note that i a free group, give a reduced word g, the legth [g] of its cojugacy class equals the miimum legth of the cyclic permutatios of g. Let ν be the uiform distributio o the set S of reduced words of legth, ad λ be the uiform distributio o the set J. A short coutig argumet shows the followig: Lemma 2 [3], Lemma 4.1. Let p : J F be the map from cyclically reduced words to cojugacy classes. The where T V Some basic probability. p λ µ T V 0 as, deotes the total variatio of a measure. Lemma 3. Let A be ay sequece of measurable sets i a probability space, P a sequece of probability measures, ad let B a sequece such that The Proof. By elemetary set theory, lim P B = 1. lim sup P A P A B = 0. P A = P A B + P A \ B P A B + P B c which yields the claim. Lemma 4. Let P be a sequece of probability measures o a Borel space X, ad let F, G : X R be two measurable fuctios. Suppose that P Gx ɛ 0 for ay ɛ > 0, ad let c be a sequece of positive real umbers with lim c = 1. Suppose that there exists a cotiuous fuctio ρ : R R + such that The lim P F x [a, b] = b F x + Gx lim P [a, b] = c a ρx dx. b a ρx dx. Proof. Fix ɛ > 0, ad deote Φ a,b := b a ρx dx. If is sufficietly large, the by settig Y := F +G c we have {F x [a + ɛ, b ɛ] ad Gx ɛ/2} {Y [a, b]}

3 A CENTRAL LIMIT THEOREM FOR RANDOM CLOSED GEODESICS 3 hece usig Lemma 3 lim if P Y [a, b] lim if P F [a + ɛ, b ɛ] ad G ɛ/2 = = lim if P F [a + ɛ, b ɛ] = Φ a+ɛ,b ɛ. O the other had {Y [a, b] ad G ɛ/2} {F [a ɛ, b + ɛ]} hece lim sup P Y [a, b] = lim sup P Y [a, b] ad G ɛ/2 ad takig ɛ 0 completes the proof. lim sup P F [a ɛ, b + ɛ] = Φ a ɛ,b+ɛ The cetral limit theorem for displacemet. The proof of Theorem 1 uses equatio 1: basically, oe proves a CLT for the fuctio dx, gx, ad the shows that the term 2gx, g 1 x x teds to zero. These two facts will suffice by Lemma 4. We will start by establishig the CLT for displacemet. We will use the otatio N a,b := 1 b x2 2π a e 2 dx. Theorem 5. There exist costats L > 0 ad σ > 0 such that for ay a, b R with a < b we have dgx, x L ν σ [a, b] N a,b. This theorem follows promptly from [1, Theorem 3.7.6]. First of all, let us ote that the set of reduced words i the free group is i bijectio with the set of paths i a certai directed graph Γ begiig at a distiguished vertex. Thus, oe ca defie a Markov chai with iitial distributio cocetrated at this vertex. For the details i the case of a geeral hyperbolic group, see [1, 2, 5]. I the particular case of the free group, the graph Γ has a iitial distiguished vertex v 0 ad a vertex for each elemet of S S 1. There is a directed edge from v 0 to each other vertex, ad a directed edge from ay v S S 1 to ay w S S 1 as log as v w 1. All edges are labeled by their termial vertices. Now, if for each vertex we assig the same probability to all the outward poitig edges, oe sees directly from this costructio that the distributio of the th step of the correspodig Markov chai is equal to the uiform distributio o the set of reduced words of legth. Thus, the measure ν is the th -step distributio of the Markov chai, ad oe ca apply thermodyamic formalism as i Calegari [1, Sectio 3.7]. Oe just eeds to verify the Hölder cotiuity assumptios of the observable. I particular, we eed the followig: Lemma 6 [8], Propositio 1; [9], Lemma 1 ad Propositio 3. Let X be a CAT-k metric space, with k > 0, ad let x, y i X. The there exist costats C > 0 ad a > 1 such that for ay z, w X x, z y x, w y Ca z,wx.

4 4 I. GEKHTMAN, S.J. TAYLOR, AND G. TIOZZO The we apply the lemma with z = hx, w = gx, y = s 1 x ad get otig that we eed oly fiitely may values of s x, hx s 1 x x, gx s 1 x Ca gx,hxx for ay h, g G, ay s S. Sice the actio G H 2 is covex cocompact, the quatities gx, hx x ad g, h the Gromov product i the group G are coarsely equal. This shows that we ca apply the followig theorem: Theorem 7 [1], Theorem Let G be a hyperbolic group, ad S a fiite geeratig set for G. Let F : G R be a real-valued fuctio which is Lipschitz i both the left- ad right-ivariat word metrics o G, ad satisfies D s F g D s F h Ca g h for all s i S ad all g, h G, where C > 0, a > 1, ad D s F = F g F sg. The there are costats L ad σ such that if g is the th step of the Markov chai, there is covergece i probability 1/2 F g L N0, σ. Sice F = dgx, x is Lipschitz i the word metric, ad satisfies the Hölder cotiuity property as above, this establishes Theorem 5. Remark 8. Let us highlight that i geeral for hyperbolic groups oe ca costruct such a Markov chai ad Theorem 7 holds. However, what is special i the free group is that the distributio of the th step of the Markov chai is exactly equal to the uiform distributio o the set of elemets of word legth i the Cayley graph. I geeral, the two measures are equivalet up to multiplicative costats, which makes it harder to traslate the CLT from oe settig to the other. Now, we use the previous Theorem which is about the Markov chai to study the distributio of cyclically reduced words. Give a reduced word g of legth, let ĝ deote the prefix of g of legth log. Let λ be the uiform distributio o all cyclically reduced words of legth. Let λ,m deote the distributio of the prefix of legth m of a uiformly chose cyclically reduced word of legth. That is, if g has law λ, the ĝ has law λ,log. We defie ν,m i the same way, usig ν i place of λ. By the Markov property, ν,m = ν m. We eed the followig lemma, which is stated for a free group of rak N. Lemma 9 [3], Lemma 4.5. For ay 1 m 1, 1 + 2Nθ m λ,m ν,m T V 2 1 2Nθ m 1 where θ = 1 2N 1. Theorem 10. For ay a, b R with a < b oe has dgx, x L λ σ [a, b] N a,b as.

5 A CENTRAL LIMIT THEOREM FOR RANDOM CLOSED GEODESICS 5 Proof. Sice log / 0, Theorem 5 together with Lemma 4 implies dgx, x L ν log σ [a, b] N a,b. Now, sice ν log = ν,log ad by Lemma 9 we get hece λ,log ad by the defiitio of ĝ λ dĝx, x L σ ν,log λ,log T V 0 dgx, x L σ [a, b] = λ,log g : [a, b] N a,b, dgx, x L σ Fially, ote the sice the orbit map G H 2 is Lipschitz, we have dgx, x dĝx, x C log where C is the Lipschitz costat. The usig Lemma 4 lim λ dgx, x L g : σ [a, b] = lim λ g : which completes the proof. = N a,b dĝx, x L σ [a, b]. [a, b] The Gromov product. The remaiig step of our proof is to tur the statemet about displacemet Theorem 10 ito a statemet about traslatio legth. This is doe by cotrollig the Gromov product. Propositio 11. Let λ deote the uiform distributio o the set of cyclically reduced words of legth. The for ay ɛ > 0, λ g : gx, g 1 x x ɛ 0. Proof. We have the followig simple estimate: if N is the rak of the free group, the for ay 2N 2 2N 1 #S #J #S. I fact, clearly J S. Moreover, #S = 2N 1#S 1 sice oe ca cotiue a word of legth 1 i exactly 2N 1 ways, while #J 2N 2#S 1 sice oe eeds to avoid both the iverse of the first letter of ad of the last letter, thus remaiig with at least 2N 2 choices to cotiue the word. The, by [[5], Propositio 7.10] we have hece ν g : gx, g 1 x x ɛ 0 λ g : gx, g 1 x x ɛ #S #J ν g : gx, g 1 x x ɛ 0 sice #S #J 2N 1 2N 2.

6 6 I. GEKHTMAN, S.J. TAYLOR, AND G. TIOZZO Proof of Theorem 1. Proof of Theorem 1. Let d := dx,gx L σ, t := τg L σ, ad p := t d = 2gx,g 1 x x+oδ σ. By the CLT for displacemet Theorem 10, for ay a < b λ d [a, b] N a,b. Moreover, by decay of Gromov products for ay ɛ > 0 Propositio 11 we have hece by Lemma 4 λ Fially, by Lemma 2 this implies which completes the proof. µ γ : λ p ɛ 0 τg L σ τγ L σ Refereces [a, b] N a,b. [a, b] N a,b [1] Day Calegari, The ergodic theory of hyperbolic groups, Cotemp. Math , [2] Day Calegari ad Koji Fujiwara, Combable fuctios, quasimorphisms, ad the cetral limit theorem, Ergodic Theory ad Dyamical Systems 30, o , [3] Moira Chas ad Steve P. Lalley, Self-itersectios i combiatorial topology: statistical structure, Ivet. Math. 188, o , [4] Moira Chas, Kere Li, ad Berard Maskit, Experimets suggestig that the distributio of the hyperbolic legth of closed geodesics samplig by word legth is Gaussia, Experimetal Mathematics 22, o , [5] Ilya Gekhtma, Samuel J. Taylor, ad Giulio Tiozzo, Coutig loxodromics for hyperbolic actios, Joural of Topology 11, o , [6] Joseph Maher ad Giulio Tiozzo, Radom walks o weakly hyperbolic groups, J. Reie Agew. Math. published olie, to appear i prit, [7] Peter S. Park, Probability laws for the distributio of geometric legths whe samplig by a radom walk i a Fuchsia fudametal group, Arxiv preprit, arxiv: [math.gt]. [8] Mark Pollicott ad Richard Sharp, Compariso theorems ad orbit coutig i hyperbolic geometry, Trasactios of the AMS 350, o , [9] Mark Pollicott, ad Richard Sharp, Poicaré series ad compariso theorems for variable egative curvature, i Topology, ergodic theory, real algebraic geometry, Amer. Math. Soc. Trasl. Ser. 2, , , Amer. Math. Soc., Providece, RI. Departmet of Mathematics, Uiversity of Toroto, 40 St George St, Toroto, ON, Caada, address: ilyagekh@gmail.com Departmet of Mathematics, Temple Uiversity, 1805 North Broad Street Philadelphia, PA 19122, U.S.A, address: samuel.taylor@temple.edu Departmet of Mathematics, Uiversity of Toroto, 40 St George St, Toroto, ON, Caada, address: tiozzo@math.toroto.edu