DISCRETE LOGARITHMS IN FREE GROUPS YIANNIS N. PETRIDIS AND MORTEN S. RISAGER. (Communicated by Wen-Ching Winnie Li)

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1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volue 134, Nuber 4, Pages S (05) Article electronically published on October 5, 2005 DISCRETE LOGARITHMS IN FREE GROUPS YIANNIS N. PETRIDIS AND MORTEN S. RISAGER (Counicated by Wen-Ching Winnie Li) Abstract. For the free group on n generators we prove that the discrete logarith is distributed according to the standard Gaussian when the logarith is renoralized appropriately. 1. Introduction Let Γ = F (A 1,...,A n ), n 2,bethefreegrouponn generators. We define additive hooorphiss on the generators log (1.1) j : Γ Z A i δ ij. Hence log j counts the nuber of occurrences (with signs) of the generator A j.we want to study the distribution of this discrete logarith as the word length grows to infinity. Our ain result is that, after a suitable re-noralization and restriction, the discrete logarith is distributed according to a standard Gaussian distribution. More precisely, let n 1 (1.2) log j (γ) = wl (γ) log j(γ). Here wl (γ) denotes the word length of γ Γ. Let Γ c be the set of cyclically reduced words in Γ. Theore 1.1. In Γ c the function log j has asyptotically a standard Gaussian distribution. More precisely #{γ Γ c wl (γ) l, log j (γ) [a, b]} 1 b ) exp ( x2 dx as l. #{γ Γ c wl (γ) l} 2π a 2 After proving this theore we discovered that I. Rivin arrived at a siilar theore in [10, Theore 5.1] (see also [12]). Rivin s proof uses certain results on Chebyshev polynoials. Our proof uses character perturbations of the adjacency operator of a singleton graph (see Figure 1 where n =4). There is a fundaental identity, due to Ihara, which relates geoetric entities (lengths of closed paths on the graph) to the spectru of the adjacency operator of the graph. We consider this identity when the syste transfors according Received by the editors August 9, 2004 and, in revised for, Noveber 12, Matheatics Subject Classification. Priary 05C25; Secondary 11M36. The first author was partially supported by PSC CUNY Research Award, No , and NSF grant DMS c 2005 Aerican Matheatical Society

2 1004 YIANNIS N. PETRIDIS AND MORTEN S. RISAGER Figure 1. to a certain sooth faily of characters related to the discrete logariths. By differentiating in the character faily we can extract enough inforation about the relevant generating functions to calculate the oents of the rando variable in Theore 1.1. Apart fro the use of this identity our proof is eleentary. Our ethod is very powerful and has been used by the authors in a series of papers [6, 7, 9, 8] to prove distribution results of additive hooorphiss in various different contexts. In fact we were otivated by our previous results to investigate the noral distribution on the level of the group without considering its action as isoetries on hyperbolic space. 2. Fro free groups to graphs Let Γ = F (A 1,...,A n ), n 2,bethefreegrouponn generators. We want to relate a finite graph to Γ. We refer to [11, 13, 4, 3] for the basics on graphs and further explanation of soe of the terinology that we use. We ostly follow the terinology of [3]. Every g Γ is uniquely represented by a reduced word, i.e. k γ = A ɛ i i, i {1,...,n},ɛ i {±1}, in which there does not exist i 0 such that i0 = i0 +1 and ɛ i0 ɛ i0 +1 = 1. Hence (2.1) log j (γ) = i =j counts (with signs) the nuber of occurrences of A j in the reduced word representation of γ. Wedefinethewordlengthofγ to be wl (γ) =k. We also define g Γ to be cyclically reduced if γ γ = k A ɛ i i ɛ i k A ɛ i i is reduced (or equivalently if either ɛ 1 ɛ k 1 or 1 k ). We consider the set Γ c of cyclically reduced words Γ c = {γ Γ γ is cyclically reduced}. There is an infinite-to-one ap ψ (cyclic reduction) fro Γ to Γ c,andthisap satisfies log j (γ) =log j (ψ(γ)). Let S = {A 1,...,A n } and consider the 2n-regular Cayley graph X = X(Γ,S), i.e. the graph having vertices Γ and (positive) edges Γ S where the edge (g, s) Γ S has the following origin and terinus: o(g, s) =g, t(g, s) =gs.

3 DISCRETE LOGARITHMS IN FREE GROUPS 1005 The group Γ acts (strictly hyperbolic) by isoetric isoorphiss on X, andthe quotient graph Γ\X is 2n-regular and has one vertex. We denote by C red the set of closed reduced paths on the graph. The set of eleents in C red of length n is denoted by C red n. For this particular graph the following is easy to verify: Lea 2.1. There is a one-to-one correspondence between C red and the set of cyclically reduced group eleents γ Γ c with wl (γ) =. We can now therefore define an additive hooorphis log j on C red by fixing an isoorphis φ :C red 1 {γ Γ c wl (γ) =1} and defining log j (C) =log j (φ(c)). By an obvious abuse of notation we shall often write log j (C) instead of log j (C). 3. The Ihara zeta function We now briefly explain the basics of the Ihara zeta function. There is a close reseblance between the Ihara zeta function and the Selberg zeta function [1]. We refer to [13, 4, 3] for further details. Let X be a (q + 1)-regular infinite tree (q 2) and let Γ be a strictly hyperbolic free subgroup of isoetric autoorphiss of X with finite (q + 1)-regular quotient graph (3.1) # vert (Γ\X) <. Furtherore let χ :Γ S 1 be a unitary character. Consider L 2 (X, χ) ={f :vert(x) C f(γv) =χ(γ)f(v) for all γ Γ,v vert (X)} together with the inner product f,g = f(v)g(v). v vert(γ\x) We note that this is a finite-diensional Hilbert space of diension di L 2 (X, χ) = #vert(γ\x). We consider the adjacency operator A(Γ,χ):L 2 (X, χ) L 2 (X, χ) defined by (3.2) (A(Γ,χ)f)(v) = v v f(v ). Here v v eans that v is adjacent to v. The adjacency operator is self-adjoint with spectru contained in [ (q +1),q+1]. Let P Γ be the set of priitive conjugacy classes in Γ different fro the unit class. For {P } P Γ we define the degree (3.3) deg(p )= in l(v, Pv) v vert(x) where l(v, v ) is the length of the geodesic fro v to v. The Ihara zeta function is defined, for u C with u <q 1,by (3.4) Z(Γ,χ; u) = (1 χ(p )u deg P ) 1. {P } P Γ There is a fundaental identity, due to Ihara, which relates this geoetric object to the spectru of the adjacency operator. More precisely, we have the following explicit deterinant representation of the Ihara zeta function [4].

4 1006 YIANNIS N. PETRIDIS AND MORTEN S. RISAGER Theore 3.1 (Ihara). (3.5) Z(Γ,χ; u) =(1 u 2 ) g det (1 ua(γ,χ)+qu 2 ) 1, where g =(q 1)# vert (Γ\X) /2. This gives iediately that the Ihara zeta function adits eroorphic continuation to the whole coplex plane. We consider the logarithic derivative of Z(Γ,χ; u) and find u d χ(p log Z(Γ,χ; u) = u )degpudeg P 1 (3.6) du 1 χ(p )u deg P {P } P Γ = χ(p n )degpu n deg P n=1 {P } P Γ = n Γ,χ ()u, where (3.7) n Γ,χ () = χ(p n deg P = n N,{P } P Γ n )degp. There is a one-to-one correspondence between priitive conjugacy classes of Γ of degree d, P Γ (d) and priitive cycles in Γ\X of length d, Pri Γ\X (d). The natural ap fro the set of closed reduced prie paths of length d in Γ\X, C red,prie d,to the set of priitive cycles of length d in Γ\X is d-to-one (each of the d vertices can be a starting point for the cycle). We therefore have d χ(c /d )= χ(c). n Γ,χ () = d {P } P Γ χ(p /d )= d,prie d Using (3.5) we get (3.8) n Γ,χ ()u = 2gu2 d 1 u 2 u du det(1 ua(γ,χ)+qu2 ) det(1 ua(γ,χ)+qu 2 ). When χ = 1 the adjacency operator has q + 1 as an eigenvalue with the identity as eigenfunction. Hence, in this case, we see that (3.8) has a pole at u = q Generating functions We now explain how to use the Ihara zeta function to get inforation about the generating functions relevant to Theore 1.1. Since we are ultiately interested in the case of the singleton graph constructed in section 2 we assue for siplicity that the quotient in the previous section has only one vertex. In the case of the graph fro section 2 we have q +1=2n. We define a faily of unitary characters on Γ by χ ɛ (γ) =exp(iɛ log j (γ)). In this siple situation the adjacency operator is siply a ultiplication operator: n (4.1) (A(Γ,χ ɛ )f)(v) = (χ ɛ (A i )+χ ɛ (A i ) 1 )f(v).

5 DISCRETE LOGARITHMS IN FREE GROUPS 1007 For siplicity we let (4.2) A(ɛ) = n (χ ɛ (A i )+χ ɛ (A i ) 1 )=2(n 1) + 2 cos(ɛ). Equation (3.8) now takes the for (4.3) n Γ,χɛ ()u = 2gu2 ua(ɛ) 2qu2 + 1 u2 1 ua(ɛ)+qu 2. We denote the left-hand side by G(u, ɛ). This is the generating function relevant to the proble we are studying. When ɛ = 0 this function has a siple pole at u = q 1 with residue (4.4) res u=q 1 G(u, 0) = q 1. We wish to understand (4.5) G (k) (u) := dk G(u, ɛ) dɛk = ɛ=0 i k log j (C) k u. The convergence is unifor in ɛ, by coparison with the series with ɛ =0,sowe can differentiate terwise. Fro (4.3) we get iediately the following result: Theore 4.1. The function G (k) (u) adits eroorphic continuation to the whole coplex plane. The points u = q 1, u =1are the only possible poles when n 1. The function G(u) =G (0) (u) has poles at u = ±1 and u = q 1. We now calculate the pole order and leading ter of the pole at u = q 1.When k 1, Eq. (4.3) gives ( k (4.6) G (k) (u) = )(ua (l) (0) δ l=0 2qu 2 ) dk l l dɛ k l (1 ua(ɛ)+qu2 ) 1. ɛ=0 l=0 Clearly we have 2n, if l =0, (4.7) A (l) (0) = 0, if l is odd, 2( 1) l/2, if l 2iseven. We first analyze the leading ter L l of d l (4.8) dɛ l (1 ua(ɛ)+qu2 ) 1. ɛ=0 Since (4.9) (1 ua(ɛ)+qu 2 )(1 ua(ɛ)+qu 2 ) 1 =1, we find that (4.8) equals l 1 ( l (4.10) (1 ua(0) + qu 2 ) )A 1 (l r) (0)u dr r dɛ r (1 ua(ɛ)+qu2 ) 1. ɛ=0 r=0

6 1008 YIANNIS N. PETRIDIS AND MORTEN S. RISAGER When l = 1 it is obvious fro (4.7) that this vanishes. By induction and (4.7) we conclude that d l (4.11) dɛ l (1 ua(ɛ)+qu2 ) 1 =0 ɛ=0 when l is odd. Alternatively 1 ua(ɛ)+qu 2 is an even function. Fro (4.10), (4.11), and (4.7) an easy induction arguent now shows that when l is even (4.8) has a pole of order l/2 + 1 and the leading ter L l satisfies the recurrence relation ( ) l (4.12) L l =(res u=q 1(1 A(0)u + qu 2 ) 1 ) A (2) (0)q 1 L l 2. l 2 It follows easily that (4.13) L l = 1 l! q l/2 (q 1). l/2+1 Therefore, the ain ter in G (k) (u), see (4.6), coes fro l = 0, and using (4.13) we get the following theore: Theore 4.2. When k is odd the function G (k) (u) is identically zero. When k is even G (k) (u) has a pole at u = q 1 of order k/2 +1 and the leading ter in the Laurent expansion around u = q 1 equals k! q k/2+1 (q 1). k/2 We notice that the vanishing of G (k) (u) fork odd is also clear fro the fact that log j (C) k =0. This fact is clear since if C is a reduced path of length, then the inverse path is also a reduced path of length and log j (C) k +log j (C 1 ) k =0when is odd. 5. Suations In this section we want to find asyptotics as T of the following sus: (5.1) log j (γ) k. γ Γ c wl(γ) T In principle we can find exact expressions for these sus. We show how this is done when k = 0. In the general situation we settle with asyptotics since this is all we need for Theore 1.1. We do this on the basis of the calculations of the previous chapter and a Tauberian theore to be proved below. We notice that fro Lea 2.1 and the fact that if φ(c) =γ, thenl(c) =wl(γ) wehavethat(5.1)equals (5.2) log j (C) k. T For k odd the discussion at the end of the last section shows that (5.1) is identically zero. We have 1 1 u2n + qu 2 = 1 ( q q 1 1 qu 1 ). 1 u

7 DISCRETE LOGARITHMS IN FREE GROUPS 1009 Fro this we see, using a geoetric expansion, that u2n 2qu 2 1 u2n + qu 2 = (q +1)u. Since we conclude that G (0) (u) = 2gu 2 1 u 2 = q 1 2 #C red u = (1 + ( 1) ) u, ( q +1+ q 1 ) (1 + ( 1) ) u 2 when u <q 1. Using Lea 2.1 and the uniqueness of Taylor expansions, we proved the following theore: Theore 5.1. Let Γ be the free group on n eleents and Γ c the set of cyclically reduced words in Γ. Then the nuber of cyclically reduced words of word length equals #{γ Γ c wl (γ) =} =(2n 1) +1+(n 1) (1 + ( 1) ). We notice that this is [10, Theore 1.1]. In principle we can use the sae technique as above for any k using (4.6). This would give explicit, though quite coplicated, expressions for log j (C) k. For siplicity we use only the fact that we have calculated the leading ter of (4.6) to obtain the ain ter and sharp error estiates for (5.2). To do this we need the following proposition: Proposition 5.2. Assue that f(u) = c u =0 has radius of convergence R = q 1 1. Assue further that (i) f adits eroorphic continuation to an open set containing the closed unit disc in C; (ii) the points u = q 1 and u =1are the only possible poles in the closed unit disc; (iii) the pole at q 1 is of order k with leading ter a k (u q 1 ) k. Then l ( q) k a k c = (q 1)(k 1)! ql+1 l k 1 + O(q l+1 l k 2 ). =0 To prove this proposition we need the following eleentary lea. We shall write (k) = k(k +1) (k =( 1)).

8 1010 YIANNIS N. PETRIDIS AND MORTEN S. RISAGER Lea 5.3. Let q>1 and k N. Then (5.3) (5.4) l =0 l =0 (k)! (k)! = 1 k! lk + O(l k 1 ), q 1 = (q 1)(k 1)! ql+1 l k 1 + { O(q l+1 l k 2 ), if k 2, O(1) if k =1. Proof. To prove (5.3) we notice that (k) /! =(+1)(+2) (+k 1)/(k 1)!. Using the eleentary estiate l k 1 = 1 k lk + O(l k 1 ) proves the clai. To prove (5.4) we define =0 a(k,, q) = (k) q and A(k, l, q) =! l a(k,, q). Then using a(k,, q) =(k + 1)a(k 1,,q)/(k 1) (when k 1) and partial suation we find A(k, l, q) = 1 l a(k 1,,q)(k + 1) k 1 =0 = 1 1 (5.5) A(k 1,l,q)(k + l 1) A(k 1,t,q)dt k 1 k 1 0 = l 1 l A(k 1,l,q)+A(k 1,l,q) A(k 1,t,q)dt. k 1 k 1 0 When k = 1 the clai is obvious since in this case (k) /! = 1. The reaining cases now follow fro (5.5) by induction. To prove Proposition 5.2 we let k a j (u q 1 ) j, resp. =0 l b j (u 1) j, be the singular parts of f at u = q 1,resp.u = 1. Now the function (5.6) f(u) k a j (u q 1 ) j b j (u 1) j is regular in an open disc containing the closure of the unit disc. Using the series expansions 1 (u 1) j = ( 1) j (j)! u, 1 (u q 1 ) j = q j ( 1) j =0 (j) q u,! =0

9 DISCRETE LOGARITHMS IN FREE GROUPS 1011 we conclude that the series c a j ( q) j (j)! q =0 k b j ( 1) j (j)! u has radius of convergence strictly larger than 1. In particular it is convergent at u =1,thatis (5.7) l c = =0 l =0 a j ( q) j (j)! q + l k =0 b j ( 1) j (j)! as l. It follows now fro Lea 5.3 that l ( q) k a k (5.8) c = (q 1)(k 1)! ql+1 l k 1 + O(q l+1 l k 2 ), =0 + o(1) which finishes the proof of the proposition. We notice that we have proved a Wiener-Ikehara-type Tauberian theore (see [2]) for power series with general coefficients under the assuption that the su adits eroorphic continuation to a disk containing the closure of the unit disc. Obviously the proposition can be easily generalized in any directions, but since we only need it in the generality stated we refer fro doing so. Using Proposition 5.2, Theore 4.1, and Theore 4.2 we get Theore 5.4. l { 0 if k is odd, log j (C) k = k! 1 (k/2)! q l+1 l k/2 + O(q l+1 l k/2 1 ) (q 1) k/2+1 if k is even. =0 We now define the noralized discrete logariths to be g log j (C) = l(c) log j(c). Here l(c) is the length of the cycle C. We then define the rando variable X l with probability easure P (X l [a, b]) = #{C Cred l(c) l, log j (C) [a, b]} #{C C red. l(c) l} We then calculate the asyptotic oents of these rando variables, i.e. we find the asyptotics of 1 M k (X l )= #{C C red log j (C) k. l(c) l} l(c) l Fro Theore 5.4 we find, using partial suation, that as l, 0 if k is odd, (5.9) M k (X l ) k! if k is even. 2 k/2 (k/2)!

10 1012 YIANNIS N. PETRIDIS AND MORTEN S. RISAGER The left-hand side equals the oents of the standard Gaussian and fro a classical result due to Fréchet and Shohat (see [5, 11.4.C]) we conclude that (5.10) P (X l [a, b]) 1 b ) exp ( x2 dx as l. 2π a 2 Using the identification in Lea 2.1 and the fact that if φ(c) =γ, thenl(c) = wl (γ) we arrive at Theore 1.1. Acknowledgents We would like to thank Alexei B. Venkov, Dorian Goldfeld and Søren Galatius for valuable coents and suggestions. References [1] D. Hejhal, The Selberg trace forula for PSL(2, R). Vol. 1. Lecture Notes in Matheatics, Springer-Verlag, Berlin, 1976, vi+516pp. MR (55:12641) [2] J. Korevaar, A century of coplex Tauberian theory. Bull. Aer. Math. Soc. (N.S.) 39 (2002), no. 4, (electronic). MR (2003g:40004) [3] A. M. Nikitin, The Ihara-Selberg zeta function of a finite graph and sybolic dynaics, Algebra i Analiz 13 (2001), no. 5, ; translation in St. Petersburg Math. J. 13 (2002), no. 5, MR (2003f:11137) [4] A. M. Nikitin and A. B. Venkov, The Selberg trace forula, Raanujan graphs and soe probles in atheatical physics. (Russian), Algebra i Analiz 5 (1993), no. 3, 1 76; translation in St. Petersburg Math. J. 5 (1994), no. 3, MR (94:11066) [5] M. Loève, Probability theory. I, Fourth edition, Springer, New York, MR (58:31324a) [6] Y. N. Petridis, Spectral deforations and Eisenstein Series Associated with Modular Sybols. Internat. Math. Res. Notices 2002, no. 19, MR (2003h:11057) [7] Y. N. Petridis, M. S. Risager, Modular sybols have a noral distribution, Geo. Funct. Anal. 14 (2004), no. 5, MR [8] Y. N. Petridis, M. S. Risager, The distribution of values of the Poincaré pairing for hyperbolic Rieann surfaces, J. Reine Ang. Mat. 579 (2005), [9] M. S. Risager, On the distribution of odular sybols for copact surfaces, Internat. Math. Res. Notices 2004, No. 41, MR [10] I. Rivin, Growth in free groups (and other stories), arxiv:ath.co/ [11] J.-P. Serre, Trees, Translated fro the French original by John Stillwell, Corrected 2nd printing of the 1980 English translation, Springer, Berlin, MR (2003:20032) [12] R. Sharp, Local liit theores for free groups, J. Math. Ann. 321 (2001), 4, p MR (2002k:20039) [13] A. Terras, Fourier analysis on finite groups and applications, Cabridge Univ. Press, Cabridge, MR (2000d:11003) Departent of Matheatics and Coputer Science, City University of New York, Lehan College, 250 Bedford Park Boulevard, West Bronx, New York Current address: The Graduate Center, Matheatics Ph.D. Progra, 365 Fifth Avenue, Roo 4208 New York, New York E-ail address: petridis@coet.lehan.cuny.edu Departent of Matheatical Sciences, University of Aarhus, Ny Munkegade Building 530, 8000 Aarhus, Denark E-ail address: risager@if.au.dk