KLEINIAN GROUPS AND HOLOMORPHIC DYNAMICS

International Journal of Bifurcation and Chaos, Vol. 3, No. 7 (2003) 959 967 c World Scientific Publishing Company KLEINIAN GROUPS AND HOLOMORPHIC DYNAMICS C. CORREIA RAMOS Departamento de Matemática and CIMA, Universidade de Évora, Portugal ccr@uevora.pt J. SOUSA RAMOS Departamento de Matemática, Instituto Superior Técnico, Lisboa, Portugal sramos@math.ist.utl.pt Received January 20, 200; Revised December 5, 200 It is known as a correspondence between iteration of rational maps and Kleinian groups, and is usually designated as the Sullivan s dictionary. This dictionary enumerates analogies between iteration of holomorphic endomorphisms and action of Kleinian groups. We propose to study explicitly examples establishing the correspondence between the two theories: iteration theory and Kleinian groups. Keywords: Kleinian groups; holomorphic dynamics; Hecke groups; iteration theory. In this work we explore some relations between iteration theory and theory of Kleinian groups. There is already work done in that direction. In the case of Fuchsian groups there is the result of Bowen and Series [979] which guarantees the existence of a boundary map, orbit equivalent to the action of the Fuchsian group on the hyperbolic plane. We use this result in Sec. 3. Brooks and Matelski [98] used Kleinian groups and complex iteration in relation with the question of deciding when a given twogenerator group is discrete, using the Jorgensen inequality. This approach is also followed by Gehring and Martin [989]. On the other hand, the existence is known of many analogies between iteration theory of holomorphic transformations and the theory of Kleinian groups. These analogies are collected in what is usually called Sullivan s dictionary. The number of these analogies and the importance of the results suggest deep connections between them. We present, in Sec. 2, part of the Sullivan s dictionary. We also present some known results about Kleinian groups which allow us to give explicit examples of relations between iteration theory and Kleinian groups. In Sec. 3 we consider a particular kind of Kleinian groups, the Fuchsian groups, and using the result in [Bowen & Series, 979], we give explicit relations between these groups and the iteration of piecewise fractional linear transformations. We build subshifts associated to Hecke groups, a one-parameter family of Fuchsian groups. In Theorem 3.3 to every Hecke group H q, we associate finite type subshift defined by a transition matrix with dimension q 2. In Theorem 3.4 and Corollary 3.5 we characterize the topological entropy of the subshifts associated to the Hecke groups. The methods here introduced can be generalized to other families of Fuchsian groups. Finally in Sec. 4 we study automorphisms of Kleinian groups and we use it to establish a different kind of relation between iteration theory and Kleinian groups. We focus on a one-parameter holomorphic family of Kleinian groups which 959

960 C. Correla Ramos & J. Sousa Ramos corresponds to the family of quasi-fuchsian groups introduced in [Wright, 988]. See also [Keen et al., 993]. Using iteration theory we identify where these groups are in the parameter space. In Theorem 4.2, we state that the parameters for those Kleinian groups do not belong to the Cartesian product of the Mandelbrot set and the filled Julia set of a quadratic map. Thus, a sequence of subgroups defined by iterating the commutator of the generators, have their traces given by a polynomial transformation which implies discreteness criteria for subgroups in P SL 2 (C). -. Preliminaries We can define Kleinian Group as a discrete subgroup Γ of P SL 2 (C). An element of P SL 2 (C) can be represented by a matrix of SL 2 (C) ( ) a b γ = c d Fig.. Limit set of the Kleinian group Γ a with a = 2. with ad bc =, reminding the fact that P SL 2 (C) SL 2 (C)/{±I}. The elements of P SL 2 (C) act on H 3 as orientation-preserving isometries, and act on C as automorphisms, γ(z) = (az + b)/(cz + d), Möbius transformations that preserves orientation. The set Γ(p) = {γ(p) : γ Γ} is denoted as the orbit of p H 3. The set Γ(p) has accumulation points only on the Riemann Sphere C = H 3. These points form a closed set called the limit set of Γ, Λ(Γ) (see Fig., for Γ a = F, G a ( ) ( 0 a =, I a 2 ) a 2, a with a = 2). The set Ω(Γ) = C \Λ(Γ) is called the regular set. According to the type of limit set Λ(Γ), a Kleinian group is called: Elementary, if Λ(Γ) consists of at most two points. Otherwise it is nonelementary. Fuchsian, if Γ has an invariant disc D. The elements in P SL 2 (C) can be classified according to their set of fixed points or, which is equivalent, by the values of the square of the traces for the corresponding matricial representatives: g P SL 2 (C) is parabolic (resp. elliptic, hyperbolic, loxodromic) if T r 2 (g) is 4 (resp. [0, 4[, ]4, + ], / [0, + ]). Fig. 2. Julia set of g β (z) = z( + β z) 2, with β = 2i. Now recall some basic notions on dynamics of rational maps. Given a rational application f : C C we define: The set of escaping points for f E(f) = {z C : lim f n (z) = } n + The filled Julia set K(f) = C E(f).

Kleinian Groups and Holomorphic Dynamics 96 The boundary J (f) = K(f) is called the Julia set of f (see Fig. 2). The set M(f) = {c C n : the critical points of f having a bounded orbit under f} is called the Mandelbrot set. 2. Sullivan s Dictionary and Some Known Facts about Kleinian Groups The Sullivan s dictionary consists of analogies between iteration of rational maps and action of Kleinian groups. A more complete description of this dictionary can be found in [Sullivan, 985]. We present here part of it. Complex Dynamics f : C C rational map of degree > 2, Julia set J (f), Fatou set F(f), Mandelbrot set M(f), J (f) : smallest closed nonempty invariant set, #J (f) 3, repelling periodic points are dense in J (f), #{components of F(f)} = 0,, 2, or, No wandering domains theorem: Every f rational with a finite number of singular values has no wandering domain, degree k gives 2k 2 complex parameters. Kleinian Groups Γ Kleinian group, nonelementary and finitely generated, Limit set Λ(Γ), Regular set Ω(Γ), Parameter space Par(Γ), Λ(Γ) : smallest closed nonempty invariant set, #Λ(Γ) 3, loxodromic fixed points are dense in Λ(Γ), #{components of Ω(Γ)} = 0,, 2, or, Ahlfors finiteness theorem: For a finitely generated Kleinian group Γ, Ω(Γ)/Γ is a finite union of analytically finite Riemann Surfaces, n generators give 3n 3 complex parameters. In what follows we present some results that are relevant in the theory of Kleinian groups and which are useful for the objectives of this work. They can be found, for example, in [Beardon, 983]. Concerning the question of deciding if a given group of Möbius transformations is discrete there is an important result on groups with two generators. Theorem 2.. (Jorgensen inequality) If the group generated by A, B P SL 2 (C), A, B, is a nonelementary Kleinian group then Tr 2 (B) 4 + Tr([A, B]) 2. This gives a necessary condition for a given group to be a discrete nonelementary group. As far as we know, there is not a necessary and sufficient condition for a given group to be Kleinian. Nevertheless there is a result that tells us that it is enough to decide the question to groups with two generators. Theorem 2.2. A nonelementary subgroup of P SL 2 (C), Γ, is discrete if and only if each twogenerator subgroup of Γ is discrete. Remark 2.3. There is a recent result by Wang and Yang [2000] which establishes that a nonelementary subgroup of P SL 2 (C), Γ, is discrete if and only if each subgroup of Γ generated by two loxodromic elements is discrete. Concerning the two-generator groups, it is known that a two-generator group Γ = A, B is determined, up to conjugacy by numbers α = Tr 2 (A) 4, β = Tr 2 (B) 4 and γ = Tr([A, B]) 2 provided that γ 0 (see [Gehring et al., 997]). These values are called the parameters of the group Γ. We have then par(γ) = (Tr 2 (A) 4, Tr 2 (B) 4, Tr[A, B] 2) = (α, β, γ) C 3 It would be interesting to describe the subset of C 3 corresponding to the parameters of Kleinian two-generator groups. There is some work done in that direction, see [Gehring et al., 997; Baribeau & Ransford, 2000]. A result that is very useful when trying to prove if a given group is a nonelementary Kleinian group, and which is a step forward in the direction of iteration theory is Proposition 2.4. For A P SL 2 (C) and θ A : P SL 2 (C) P SL 2 (C) θ A (B) = BAB if a Kleinian group A, B satisfies θa n (B) = A for some integer n, and A is not of order 2 then F ix(a) is invariant under B. In particular A, B is elementary.

962 C. Correla Ramos & J. Sousa Ramos 3. Relations Between Iteration Theory and Fuchsian Groups 3.. Hecke groups In this section we give an explicit relation between iteration theory and a particular case of Kleinian groups Fuchsian groups. The possibility of associating to a Fuchsian group a map is known, defined on the boundary of H 2, R { }, that is orbit equivalent to the group [Bowen & Series, 979]. We build explicitly this map for the Hecke groups. A Hecke group, H q, is a one-parameter family of Fuchsian groups generated by the following transformations A(z) = u( z) and B(z) = z. If u < 4, the groups are discrete only for the parameter of the form u = 4 cos 2 (π/q), q integer q 3 (the case q = 3 corresponds to the modular group P SL 2 (Z)). The boundary map is built taking into account an appropriate fundamental domain for the group. There are many possible choices. We have chosen as the fundamental domain for the Hecke groups the polygon delimited by the vertices i, i+, (+i 3)/2,. In fact we build a boundary map for each group H q, so we have a one-parameter family of maps, f q. The family of piecewise fractional-linear transformations associated to the Hecke groups, are q=6 2-2 - 2 - -2 Fig. 3. Graph of f q when q = 6, and the orbit of the point 0. given by B (x) if x A (x) if < x 0 f q (x) = A (x) if 0 < x B (x) if x For each map, f q, we can build a Markov partition and the associated Markov matrix, M q. Example 3.. For the case q = 3 we have the following partition I =], ], I 2 =], 0], I 3 =]0, ] I 4 =], 2], I 42 =]2, + ] and the matrix 0 0 0 0 0 0 0 M 3 = 0 0 0 0 0 0 0 0 0 0 Example 3.2. case q = 4 We present here the matrix for the 0 0 0 0 0 0 0 0 0 0 0 0 0 0 M 4 = 0 0 0 0 0 0 0 0 0 0 0 0 In general we have the following result Theorem 3.3. To every Hecke group H q we can associate the Markov matrix 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 M q = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 with dim(m q ) = q + 2.

Kleinian Groups and Holomorphic Dynamics 963 Proof. The partition for the general case is given by the following with I = [, ] I 2 = [, 0] I 3 = [0, x ] I 32 = [x 2, x 3 ] q 2 I I 2 I 3(q 2) = [x q 3, ] I 4 = [, + x ] I 42 = [ + x, + ] i=2 I 3i I 4 I 42 = R { } and where x = /u, and x i = A(x i ), A being one of the group generators, defined above. Looking for the orbits of the critical points we can determine the possible transitions by the action of the boundary map, f q. We have f q (I ) = I I 2 f q (I 2 ) = I 42 f q (I 3i ) = I 3(i+), i =,..., q 3 f q (I 3(q 2) ) = I 4 I 42 f q (I 4 ) = I 3 f q (I 42 ) = q 2 i=2 I 3i I 4 I 42 So the matrix, that codifies these transitions is M q presented in the statement. With matrices, it is possible to determine the topological entropy that becomes an invariant for the Hecke group via topological dynamics. The topological entropy may be determined as the logarithm of the growth number, s(f q ), s = lim n n l(fq n ) where l(f n q ) is the lap number of f n q. It is known to be equal to the spectral radius of the matrix M q (see [Lind & Marcus, 995]). We present some of the growth numbers s for the groups associated with the first values of q: q s (aprox.) 3.6803 4.83929 5.92756 6.96595 7.98358 It is clear that the growth numbers approach the value 2. In fact we prove the following results Theorem 3.4. The topological entropy of the subshift associated to the Hecke groups H q is given by h t = log s, where s (spectral radius of the matrix M q ) is the greatest real zero of the polynomial p q (t) = t q t q 2 t. Proof. Consider the matrix M q ti q+2 where I q+2 is the (q + 2) (q + 2) identity matrix. This matrix is reducible and is decomposed in the following way t 0 t M q ti q+2 = 0 0 b t b,q....... 0 0 b q, b q,q t so det(m q ti q+2 ) = t(t ) det(b q ti q ). If we set N q ti q = P (B q ti q )Q, where P, Q are convenient elementary matrices, we obtain a matrix N q ( ) Cq ti q 0 N q ti q = t such that det(n q ti q ) = det(b q ti q ) and C q is the companion matrix with characteristic polynomial t q t q 2 t. Corollary 3.5. The entropies h t, associated to the subshifts M q, for the Hecke groups H q, are bounded above by log 2. Proof. Relating the polynomial p q (t) = t q t q 2 t, as q, with the geometric series, it is easy to see that the zero of p (t) is t = 2. It is relevant to note that like in the Kneading Theory [Milnor & Thurston, 988] the dynamics here is completely determined by the symbolic sequences (kneading sequences) corresponding to the

964 C. Correla Ramos & J. Sousa Ramos itineraries of the lateral limits of the discontinuous points. In this context the symbolic sequences have additional meaning than just symbolic coding, they have a direct interpretation in group words. So we may speak of kneading words in the group, and they codify topological information about the group. We have the following kneading words discontinuous point kneading word + A BA q 3 B[A q 2 B] + B A [B] 0 + [A q 2 B] 0 + A [B] + B[A q 2 B] + A[B] 3.2. Fuchsian groups generated by two parabolic transformations We now consider a different family of Fuchsian groups, those generated by the parabolic transformations A ρ (z) = z ρz +, and B ρ(z) = z + ρ where ρ R. We set G ρ = A ρ, B ρ. This group is discrete if and only if ρ = 2 cos(π/q), q 3 or ρ 2 (see [Beardon, 988]). Based on the fundamental domain determined by the region Re(z) < ρ/2, and the exterior of the circles z (/ρ) = /ρ, z + (/ρ) = /ρ, we have the following boundary map, for the case ρ =, B (x) if x 2 A (x) if 2 < x 0 f(x) = A (x) if 0 < x 2 (x) if x 2 B The Markov matrix is 0 0 0 0 0 0 0 M =. 0 0 0 0 0 0 0 To exemplify how this construction carries topological information, we present other family of Fuchsian groups, in which if we change continuously the parameters, the partition is always the same, and so the topological information is the same. The surface obtained by H 2 /G is the independent of the parameters values. The generators are with A(z) = λ 2 z, and B(z) = pz + p2 r 2 z + p r = p λ2 λ 2 +, p, λ R. The family of associated boundary maps is the following A (x) if x p r B (x) if p r < x p + r f λ,p (x) = A(x) if p + r < x p r B(x) if p r < x p + r A (x) if p + r < x The partition and consequently the Markov matrix 0 0 0 0 0 0 0 0 0 0 0 0 0 M = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 are independent of the parameters λ, p. The topological entropy associated with the Markov subshift is log 3, which indicates the fact that the group is free. See in Fig. 4 a fundamental domain and Fig. 5 the map f 2,. 2.5.5 0.5-2 - 2 Fig. 4. Fundamental domain for G λ,p = A λ,p, B λ,p, λ = 2 and p =.

Kleinian Groups and Holomorphic Dynamics 965 3 2-2 - 2-2 -3 Fig. 5. Graph of the map f 2,. 4. Some Special Automorphisms of the Kleinian Groups As was said before, given a group generated by two transformations, it is possible to assign parameters that characterize the group up to conjugacies (see [Gehring et al., 997]). We write par(γ) = (Tr 2 (A) 4, Tr 2 (B) 4, Tr[A, B] 2) = (α, β, γ) C 3 With this notation the Jorgensen inequality is expressed by β + γ. We have that if β + γ < then the group Γ is either not discrete or elementary. It is possible to explore some of the twogenerator subgroups of Γ using automorphisms in the group. If we consider the following automorphism θ B : P SL 2 (C) P SL 2 (C) θ B (A) = ABA it is possible to prove that, defining we have z n = Tr[B, θ n B A] 2, z n = z n (z n β). If we iterate f β (z) = z(z β), with the initial condition z 0 = γ, we are obtaining the subgroups Γ n with par(γ n ) = (β, β, z n ) = (β, β, z n (z n β)). Conclusions on discreteness of the group Γ can be taken from this iteration since if we know that some subgroup is not discrete then the group itself cannot be discrete. In [Brooks & Matelski, 98] and [Gehring & Martin, 989] the authors identify the Jorgensen inequality as the basin attraction of the fixed point. We consider the set (M(f β ) K(f β )), the interior of the Cartesian product of the Mandelbrot set, M(f β ) and the filled Julia set K(f β ) for the map f β (z). Elements (β, γ) (M(f β ) K(f β )) correspond to nondiscrete or elementary groups Γ with par(γ) = (α, β, γ). With the automorphism θ B only a particular path in the group is explored. It is possible to give different kinds of automorphisms with different associated polynomials. Iteration with group automorphisms corresponds to iteration of polynomials with the parameter γ as initial condition. Another example is φ B : P SL 2 (C) P SL 2 (C) φ B (A) = ABA BA The polynomial associated to φ B is g β (z) = z( + β z) 2, this time is a cubic (see Fig. 2 for an example of Julia set). The sequence of subgroups is now the following par(γ n ) = (α n, β, z n ( + β z n ) 2 ). For the automorphism φ B we have the set (M(g β ) K(g β )) associated to the map g β (z). In general we have the following relations ψ B p β (M(p β ) K(p β )) where ψ B is an group automorphism, p β is the associated polynomial, and M(p β ), K(p β ) the corresponding Mandelbrot and filled Julia set. In what follows, we consider the one-parameter family of Kleinian groups G ξ = A, B ξ, with ξ C, where the generator representatives in SL 2 (C) are given by A = ( ) 2 0 and B ξ = ( iξ ) i i 0 Note that Tr[A, B] = 2. We are going to identify where these groups are in the parameter space, i.e. we are looking for par(g ξ ). Such groups were introduced in [Wright, 988] and [Keen et al., 993] where the authors identify the cusp groups. For each ξ C such Tr(W p/q ) = ±2, G ξ corresponds to a cusp group. The words W p/q are built recursively by the following process W /0 = A W / = AB ξ W / = A B ξ W 0/ = B ξ W (p+r)/(q+s) = W p/q W r/s

966 C. Correla Ramos & J. Sousa Ramos with p q = + r/s > r/s. The traces of this group words are polynomials in the variable ξ C. The recursion on the words, and the trace identity Tr(XY ) = Tr(X)Tr(Y ) Tr(XY ) for any matrices X, Y SL 2 (C) allow a definition of an iterative procedure to explore paths in the group to determine for which ξ corresponds to a cusp group. Let c = Tr(Y ), t 0 = Tr(X) and t = Tr(XY ), we have t n+2 = ct n+ t n where t n = Tr(XY n ), X = W p/q and Y = W r/s for some rationales p/q, r/s. Remark 4.. 974]. For the trace identities see [Magnus, The parameter c depends on ξ. There is a correspondence between the set of cusp groups and the set of parameters c C, namely the values of ξ C, for which there is n N such that t n = ±2. If the condition < Im(ξ) < 2 is held, then the corresponding group is discrete. We have the following result which states that for this family the parameters are outside both (M(f β ) K(f β )) and (M(g β ) K(g β )). Theorem 4.2. Let G ξ = A, B ξ be the oneparameter family of Kleinian groups defined above. If then and par(g ξ ) = (α, β(ξ), γ) = (0, ξ 2 4, 4) (β(ξ), 4) / (M(f β ) K(f β )) (β(ξ), 4) / (M(g β ) K(g β )). Proof. First we prove the result for the map f β. Since (β(ξ), 4) C R is sufficient to prove that γ = 4 / Re K o (f β ), β. Let β R, β 0. For K o (f β ) to be connected we must have that the image, by f β, of the critical point β/2 is greater than the preimage of the fixed point β +. This condition gives (β 2 /4), which is equivalent to β 2. If we set β R, β < 0, we get a similar condition in order for K o (f β ) to be connected. In this case the condition is that the image of the critical point be greater than the preimage of the fixed point 0, which is equivalent to β 4. Then the point ( 4, 4) belongs to (M(f β ) K(f β )), and for values of β 4, we have γ 4, proving the result. For the map g β with the same arguments we see that M(f β ) R = [ 4, 2], and that the line (β, 4), intersects M(f β ) K(f β ) only on ( 4, 4), and the result follows. It is natural to think in a generalization of other group automorphisms. Let G ξ = A, B ξ as above. Let ψ B be an automorphism in G with associated polynomial p β. Then (β(ξ), 4) / (M(p β ) K(p β )). Acknowledgment The authors would like to thank the referee for his suggestions and advice in the presentation of this paper. References Baribeau, L. & Ransford, T. [2000] On the set of discrete two-generator groups, Math. Proc. Camb. Phil. Soc. 28, 245 255. Beardon, A. [983] The Geometry of Discrete Groups (Springer-Verlag). Beardon, A. [988] Fuchsian groups and nth roots of parabolic generators, Holomorphic Functions and Moduli II, Mathematical Sciences Research Institute Publications (Springer-Verlag), pp. 3 22. Beardon, A. [993] Pell s equation and two generator free Möbius groups, Bull. London Math. Soc. 25, 527 532. Bowen, R. & Series, C. [979] Markov maps associated with Fuchsian groups, I.H.E.S. Publications 50, 53 70. Brooks, R. & Matelski, P. [98] The dynamics of 2-generator subgroups of P SL 2 (C), Riemann Surfaces and Related Topics, Proc. 978 Stony Brook Conf., Annals of Mathematical Studies (Princeton University Press). Gehring, F. & Martin, G. [989] Iteration theory and inequalities for Kleinian groups, Bull. Amer. Math. Soc. 2, 57 63. Gehring, F. W., Maclachlan, G., Martin, G. & Reid, A. W. [997] Arithmeticity, discreteness and volume, Trans. Amer. Math. Soc. 349, 36 3643. Keen, L. & Series, C. [993] Pleating coordinates for Maskit embedding of the Teichmüller space of punctured tori, Topology 32, 79 749. Lind, D. & Marcus, B. [995] An Introduction to Symbolic Dynamics and Coding (Cambridge University Press).

Kleinian Groups and Holomorphic Dynamics 967 Magnus, W. [974] Non Euclidean Tessellations and Their Groups (Academic Press). Milnor, J. & Thurston, W. [988] On iterated maps of the interval, Proc. Univ. Maryland 986 987, ed. Alexander, J. C., Lecture Notes in Mathematics, Vol. 342 (Springer-Verlag), pp. 465 563. Series, C. [999] Lectures on pleating coordinates for once punctured tori, Warwick preprint 07/999. Sullivan, D. [985] Quasiconformal homeomorphisms and dynamics I. Solution of the Fatou Julia problem on wandering domains, Ann. Math. 22, 40 48. Wright, D. [988] The shape of the boundary of Maskit s embedding of Teichmüller space of once punctured tori, unpublished preprint (see [Series, 999]).