DYNAMICS OF THE AUTOMORPHISM GROUP OF THE GL(2; R)-CHARACTERS OF A ONCE-PUNCUTRED TORUS WILLIAM GOLDMAN AND GEORGE STANTCHEV Abstract. Let π be a free group of rank 2. Its outer automorphism group Out(π) acts on the space of equivalence classes of representations ρ Hom(π; SL(2; C)). Let 2 SL−(2; R) denote the subset of GL(2; R) consisting of matrices of determinant 1 and − let ISL(2; R) denote the subgroup SL(2; R) i SL−(2; R) SL(2; C). The representa- tion space Hom(π; ISL(2; R)) has four connectedq componen⊂ts, three of which consist of representations that send at least one generator of π to i SL−(2; R). We investigate the dynamics of the Out(π)-action on these components. The group Out(π) is commensurable with the group Γ of automorphisms of the polynomial κ(x; y; z) = x2 y2 + z2 + xyz 2 − − − We show that for 14 < c < 2, the action of Γ is ergodic on κ−1(c). For c < 14, the − M −1 − group Γ acts properly and freely on an open subset Ωc κ (c) and acts ergodically M ⊂ on the complement of Ωc . We construct an algorithm which determines, in polynomial 3 M time, if a point (x; y; z) R is Γ-equivalent to a point in Ωc or in its complement. Conjugacy classes of 2ISL(2; R)-representations identify with R3 via the appropriate restriction of the character map χ : Hom(π; SL(2; C)) C3 −! ξ(ρ) tr(ρ(X)) ρ η(ρ) = tr(ρ(Y )) 7−! 2ζ(ρ)3 2tr(ρ(XY ))3 where X and Y are the generators of π. Corresp4 5onding4 to the F5ricke spaces of the once-punctures Klein bottle and the once-punctured M¨obius band are Γ-invariant open subsets ΩK and ΩM respectively. We give an explicit parametrization of ΩK and ΩM as subsets of R3 and we show that ΩM κ−1(c) = ? if and only if c < 14, while ΩK κ−1(c) = ? if and only if c > 6. \ 6 − \ 6 Contents List of Figures 2 1. Introduction 4 2. Background and Motivation 7 2.1. Algebraic Generalities 7 2.2. Geometric Motivation 7 2.3. The Structure of Hom(π; G) 7 Date: July 22, 2003. 1 2 WILLIAM GOLDMAN AND GEORGE STANTCHEV 2.4. Obstruction Classes and Components of Hom(π; G) 8 2.5. SL(2; C)-character Varieties 8 2.6. SL (2; R)-character Varieties 9 2.7. Surfaces Whose Fundamental Group is Free of Rank 2 10 2.8. Mapping Class Group, and the Structure of Out(π) 12 2.9. The modular group 12 2.10. Goldman's Result on the Real SL(2)-Characters of the Punctured Torus 14 3. Fricke Spaces Within the Character Variety 15 3.1. The Fricke Space of the Punctured Klein-Bottle 15 3.2. The Fricke Space of the Punctured M¨obius Band 18 4. Classification of Characters and the τ-reduction Algorithm 24 4.1. Notation 24 ] 4.2. The level sets of κ and the Fricke space of S0;2 25 4.3. The Quadratic Reflections 26 4.4. The orbit as a binary tree 26 4.5. The τ-function 27 4.6. Growth of the τ function 38 4.7. τ-reduction for Fricke-space characters 39 4.8. τ-reduction for arbitrary characters 41 4.9. The τ-Reduction Algorithm 47 5. The Action of the Modular Group on Characters 48 References 52 List of Figures 1 Constructing non-orientable surfaces from E-pieces 11 ] 2 S1;1 as a quotient space of S0;3 16 ] 3 The Two Sheets of S0;4 as a Double Cover of S0;2 19 4 The binary trees rooted at u and Qz(u) 27 1 5 Intersections of κ− (c) with z-coordinate level sets 29 6 Projective model of hc;z0 and associated objects 33 7 Points on the Λx;y-orbit: u0, u1 = Qx(u0), u2 = Qy(u1), etc 34 8 Classification of Points With Respect to the Monotonicity of τ 35 M 1 Ω 9 The set L Ω κ− (c) inside D 36 z \ 0 \ c 10 Lines, cones, and half-planes 43 11 Points of type z¯( +), z¯(+ ), and z¯(e) 44 − − 1 12 Families of hyperbolae as projections of κ− (c) L 45 \ z 13 The terminal plane of the character u = ( 0:2; 12; 10) 47 − − ACTION OF THE MODULAR GROUP 3 14 The functions z¯min(z; c) and z¯max(z; c) 51 4 WILLIAM GOLDMAN AND GEORGE STANTCHEV 1. Introduction Let π be a free group of rank 2. Its outer automorphism group Out(π) acts on the space of equivalence classes of representations ρ Hom(π; SL(2; C)). Let SL (2; R) denote the subset of GL(2; R) consisting of matrices of2 determinant 1, and let − − SL (2; R) = A GL(2; R) det(A) = 1 f 2 j g The group SL (2; R) is isomorphic to ISL(2; R) = SL(2; R) i SL (2; R) q − and in this context we identify the two as subgroups of SL(2; C). The representation space R = Hom(π; ISL(2; R)) has four connected components indexed by the elements of 1 H (π; Z2) ∼= Z2 Z2. The three non-zero elements of Z2 Z2 correspond to the com- ponents of R consisting× of representations that send at×least one generator of π to i SL (2; R). We investigate the dynamics of the Out(π)-action on these components. The−action of Out(π) on the component of SL(2; R)-representations has been recently studied by Goldman [8] By a theorem of Fricke [4], the moduli space of SL(2; C)-representations naturally identifies with affine 3-space C3 via the character map χ : Hom(π; SL(2; C)) C3 −! ξ(ρ) tr(ρ(X)) ρ η(ρ) = tr(ρ(Y )) 7−! 2ζ(ρ)3 2tr(ρ(XY ))3 where X and Y are the generators of π. Let4 [X;5Y ] b4e the comm5utator of X and Y . In terms of the coordinate functions ξ, η and ζ, the trace tr([X; Y ]) is given by the polynomial κ(ξ; η; ζ) := ξ2 + η2 + ζ2 ξηζ 2 − − which is preserved under the action of Out(π). Moreover, the action of Out(π) on C3 is commensurable with the action of the group Γ of polynomial automorphisms of C3 which preserve κ (Horowitz [9]). Note that Γ is a finite extension of the modular group and is isomorphic to PGL(2; Z) n (Z=2 Z=2) ⊕ Let R1;1 be the component of Hom(π; ISL(2; R)) consisting of representations that send both X and Y to i SL (2; R). The restriction χ11 of the character map to R1;1 is a surjection onto X = iR −iR R. The latter is isomorphic to R3 and the restriction of 1;1 × × κ to X1;1 induces a polynomial κ (x; y; z) := κ(ix; iy; z) = x2 y2 + z2 + xyz 2 11 − − − 3 where x, y and z are the standard coordinate functions on R . If u X1;1 is such that 1 2 k(u) = 2, the fiber χ11− (u) is an SL (2; R)-conjugacy class of irreducible representations 6 ACTION OF THE MODULAR GROUP 5 in R1;1. In this context, X1;1 identifies with a component of the SL (2; R)-character variety of π. Theorem A. Let κ (x; y; z) = x2 y2 + z2 + xyz 2 and let c R. Let Γ be the 11 − − − 2 automorphism group of k11. Then 1 . For 14 c < 2, the group Γ acts ergodically on κ11− (c). − ≤ M 1 . For c < 14, the group Γ acts properly and freely on an open subset Ωc κ11− (c) − M ⊂ and acts ergodically on the complement of Ωc . Since π is a free group, representations in Hom(π; SL (2; R)), or equivalently in Hom(π; ISL(2; R)), can be realized as lifts of representations in Hom(π; PGL(2; R)) and thus can be interpreted geometrically via the identification of PGL(2; R) with the full isometry group of hyperbolic 2-space H2. More precisely, let S be a surface with fun- damental group π1(S). Let G be a semisimple Lie group. Then Hom(π1(S); G) is an analytic variety upon which G acts by conjugation. Let Hom(π1(S); G)=G be the orbit space. The G-orbits parametrize equivalence classes of flat principal G-bundles over S. If X is a space upon which G acts, Hom(π1(S); G)=G is the deformation space of flat (G; X)-bundles over S. In this context, Hom(π; PGL(2; R))=PGL(2; R) identifies with the deformation space of flat H2-bundles over a surface S whose fundamental group is free of rank 2. When ρ Hom(π; PGL(2; R)) is a discrete embedding, the holonomy group ρ(π) acts properly discon2 tinuously on the fiber H2. The quotient H2/ρ(π) is homotopy-equivalent to S and affords a hyperbolic structure induced by that of H2. If the quotient is also diffeomorphic to S we call ρ a discrete S-embedding. The set ΩS of conjugacy classes of discrete S-embeddings is open in Hom(π; PGL(2; R))=PGL(2; R) and parametrizes complete hyperbolic structures on S marked with respect to a fixed set of generators of π.