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 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 representation space Hom(π, ISL(2, ℝ)) has four connected components, 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) = −x^{2} − y^{2} + z^{2} + xyz − 2

We show that for −14 < c < 2, the action of Γ is ergodic on κ^{−1}(c). For c < −14, the group Γ acts properly and freely on an open subset Ω_{c}^{M} ⊂ κ^{−1}(c) and acts ergodically on the complement of Ω_{c}^{M}. We construct an algorithm which determines, in polynomial time, if a point (x, y, z) ∈ ℝ^{3} is Γ-equivalent to a point in Ω_{c}^{M} or in its complement.

Conjugacy classes of ISL(2, ℝ)-representations identify with ℝ^{3} via the appropriate restriction of the character map

χ : Hom(π, SL(2, ℂ)) ⟶ ℂ^{3}
ρ ⟼ [\begin{matrix}ξ(ρ) \\ η(ρ) \\ ζ(ρ) \end{matrix}] = [\begin{matrix}tr(ρ(X)) \\ tr(ρ(Y)) \\ tr(ρ(XY)) \end{matrix}]

where X and Y are the generators of π. Corresponding to the Fricke spaces of the once-punctures Klein bottle and the once-punctured Möbius band are Γ-invariant open subsets Ω^{K} and Ω^{M} respectively. We give an explicit parametrization of Ω^{K} and Ω^{M} as subsets of ℝ^{3} and we show that Ω^{M} ∩ κ^{−1}(c) ≠ ∅ if and only if c < −14, while Ω^{K} ∩ κ^{−1}(c) ≠ ∅ if and only if c > 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.

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 Möbius 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 S_{0,2}^{♯} 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 S_{1,1}^{♯} as a quotient space of S_{0,3} 16

3 The Two Sheets of S_{0,4} as a Double Cover of S_{0,2}^{♯} 19 4 The binary trees rooted at u and Q_{z}(u) 27 5 Intersections of κ^{−1}(c) with z-coordinate level sets 29 6 Projective model of h_{c,z_{0}} and associated objects 33 7 Points on the Λ_{x,y}-orbit: u_{0}, u_{1} = Q_{x}(u_{0}), u_{2} = Q_{y}(u_{1}), etc 34 8 Classification of Points With Respect to the Monotonicity of τ 35 9 The set L_{z} ∩ Ω_{0}^{M} ∩ κ^{−1}(c) inside D_{c}^{Ω} 36 10 Lines, cones, and half-planes 43 11 Points of type z̄_{(−+)}, z̄_{(+−)}, and z̄_{(e)} 44 12 Families of hyperbolae as projections of κ^{−1}(c) ∩ L_{z} 45 13 The terminal plane of the character u = (−0.2, 12, −10) 47 14 The functions z̄_{min}(z, c) and z̄_{max}(z, c) 51

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 of determinant −1, and let

SL_{±}(2, ℝ) = {A ∈ GL(2, ℝ) | det(A) = ±1}

The group SL_{±}(2, ℝ) is isomorphic to

ISL(2, ℝ) = SL(2, ℝ) ⨿ i SL_{−}(2, ℝ)

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 H^{1}(π, ℤ_{2}) ≅ ℤ_{2} × ℤ_{2}. The three non-zero elements of ℤ_{2} × ℤ_{2} correspond to the components 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 ℂ^{3} via the character map

χ : Hom(π, SL(2, ℂ)) ⟶ ℂ^{3}
ρ ⟼ [\begin{matrix}ξ(ρ) \\ η(ρ) \\ ζ(ρ) \end{matrix}] = [\begin{matrix}tr(ρ(X)) \\ tr(ρ(Y)) \\ tr(ρ(XY)) \end{matrix}]

where X and Y are the generators of π. Let [X, Y] be the commutator 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) := ξ^{2} + η^{2} + ζ^{2} − ξηζ − 2

which is preserved under the action of Out(π). Moreover, the action of Out(π) on ℂ^{3} is commensurable with the action of the group Γ of polynomial automorphisms of ℂ^{3} which preserve κ (Horowitz [9]). Note that Γ is a finite extension of the modular group and is isomorphic to

PGL(2, ℤ) ⋉ (ℤ/2 ⊕ ℤ/2)

Let ℛ_{1,1} 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 χ_{11} of the character map to ℛ_{1,1} is a surjection onto 𝒳_{1,1} = iℝ × iℝ × ℝ. The latter is isomorphic to ℝ^{3} and the restriction of κ to 𝒳_{1,1} induces a polynomial

κ_{11}(x, y, z) := κ(ix, iy, z) = −x^{2} − y^{2} + z^{2} + xyz − 2

where x, y and z are the standard coordinate functions on ℝ^{3}. If u ∈ 𝒳_{1,1} is such that k(u) ≠ 2, the fiber χ_{11}^{−1}(u) is an SL_{±}(2, ℝ)-conjugacy class of irreducible representations in ℛ_{1,1}. In this context, 𝒳_{1,1} identifies with a component of the SL_{±}(2, ℝ)-character variety of π.

Theorem A. Let κ_{11}(x, y, z) = −x^{2} − y^{2} + z^{2} + xyz − 2 and let c ∈ ℝ. Let Γ be the automorphism group of k_{11}. Then

▷ For −14 ≤ c < 2, the group Γ acts ergodically on κ_{11}^{−1}(c).

▷ For c < −14, the group Γ acts properly and freely on an open subset Ω_{c}^{M} ⊂ κ_{11}^{−1}(c) and acts ergodically on the complement of Ω_{c}^{M}.

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 H^{2}. More precisely, let S be a surface with fundamental group π_{1}(S). Let G be a semisimple Lie group. Then Hom(π_{1}(S), G) is an analytic variety upon which G acts by conjugation. Let Hom(π_{1}(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(π_{1}(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 H^{2}-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 discontinuously on the fiber H^{2}. The quotient H^{2}/ρ(π) is homotopy-equivalent to S and affords a hyperbolic structure induced by that of H^{2}. If the quotient is also diffeomorphic to S we call ρ a discrete S-embedding. The set Ω^{S} 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. We call Ω^{S} the Fricke space of S. In a certain sense Ω^{S} is a generalization of the Teichmüller space of an orientable closed surface.

Discrete embeddings inside ℛ_{1,1} give rise to non-orientable surfaces and since π is free of rank 2 the only possibilities are the once-punctured^{1} Möbius band M (equivalently, the twice-punctured projective plane), and the once-punctured Klein bottle K. Their respective Fricke spaces Ω^{M} and Ω^{K} can be parametrized as subsets of ℝ^{3}.

Theorem B. Let Γ be the group of automorphisms of the polynomial

κ_{11}(x, y, z) = −x^{2} − y^{2} + z^{2} − xyz − 2

(1) Let Ω_{0}^{M} be the region in ℝ^{3} defined by the inequalities

xy + z > 2
z < −2

1By “punctured” in this context we mean a surface with the interiors of one or more disjoint disks removed, i.e. a surface with boundary

Then the Fricke space of the once-punctured Möbius band M identifies with

Ω^{M} = ∐_{γ∈Γ/Γ_{M}} γΩ_{0}^{M}

where Γ_{M} ⊂ Γ is the stabilizer of Ω_{0}^{M}, and Γ/Γ_{M} denotes the coset space of Γ_{M}.

(2) Let Ω_{0}^{K} be the region in ℝ^{3} defined by the inequality

x^{2} + y^{2} − xyz + 4 < 0

Then the Fricke space of the once-punctured Klein bottle K identifies with

Ω^{K} = ∐_{γ∈Γ/Γ_{K}} γΩ_{0}^{K}

where Γ_{K} ⊂ Γ is the stabilizer of Ω_{0}^{K}, and Γ/Γ_{K} denotes the coset space of Γ_{K}.

Remark. The subgroups Γ_{M} and Γ_{K} correspond to the mapping class groups of M and K respectively.

The level sets κ_{11}^{−1}(c) intersect Ω^{M} if and only if c < −14, and they intersect Ω^{K} if and only if c > 6. Assume c < −14, and let Ω_{c}^{M} = κ_{11}^{−1}(c) ∩ Ω^{M}. This is precisely the region mentioned in the second part of Theorem A. In other words, on the Fricke space of the once-punctured Möbius band M the action of Γ is wandering; on the outside, the action is ergodic along the level sets κ_{11}^{−1}(c) for each c < 2. Similar statements hold for the Fricke spaces of M sitting inside the other SL_{±}(2, ℝ)-moduli-space components that correspond to non-zero classes in ℤ_{2} × ℤ_{2}.

Conjecture. Let Ω^{K} be the Fricke space of the once-punctured Klein bottle. Let Ω_{c}^{K} = κ_{11}^{−1}(c) ∩ Ω^{K}

▷ The action of Γ on Ω^{K} is wandering.

▷ For each c > 2, the action of Γ on the set κ_{11}^{−1}(c) − Ω_{c}^{K} is ergodic.

2. Background and Motivation

In this part we provide some relevant background material and interpret it in our context. We also introduce new notation and terminology that will be used in the subsequent exposition.

2.1. Algebraic Generalities. If A,B are groups, then Hom(A, B) denotes the set of homomorphisms (representations) A ⟶ B. Let Aut(A) denote the group of all automorphisms of A. There is an action of the group Aut(A)×Aut(B) on Hom(A, B) defined by

(2.1)

(Aut(A) × Aut(B)) × Hom(A, B) ⟶ Hom(A, B)
((α, β), ϕ) ⟼ β ∘ ϕ ∘ α^{−1}

In particular, the action of B on itself by conjugation embeds B in its automorphism group and thus induces an action:

\begin{matrix}B × Hom(A, B) & ⟶ Hom(A, B) \\ (b, ϕ) & ⟼ ι_{b} ∘ ϕ \end{matrix}

where ι_{b} : h ↦ bhb^{−1} is the inner automorphism of B defined by conjugation by b ∈ B. The orbit space, Hom(A, B)/B is the set of conjugacy classes of representations in Hom(A, B) and the action (2.1) descends to an action of Aut(A) on Hom(A, B)/B. Let Inn(A) denote the (normal) subgroup of Aut(A) consisting of inner automorphisms. Since Inn(A) preserves the conjugacy class of a representation, it acts trivially on Hom(A, B)/B and thus the action of Aut(A) factors through the action of the outer automorphism group

Out(A) := Aut(A)/ Inn(A)

2.2. Geometric Motivation. When A is a discrete group and B a Lie group, the representation space Hom(A, B) can have special geometric significance. In particular, it can be interpreted as the moduli space of flat bundles over manifolds. Consider, for example, a manifold M with fundamental group π = π_{1}(M, x_{0}) and a Lie group G which acts on a space X. Then the space of flat (G, X)-bundles over M can be identified with Hom(π, G)/G. In the case when M is a closed surface, and G is the group of orientation preserving isometries of the hyperbolic plane, the subset of Hom(π, G)/G consisting of equivalence classes of discrete embeddings of π, identifies with the Teichmüller space 𝔗 of M. Moreover, Out(π) is isomorphic to the mapping class group π_{0} Diff(M) of M whose action on 𝔗 is well known to be properly discontinuous.

2.3. The Structure of Hom(π, G). Whenever π is a finitely generated group, the space Hom(π, G) inherits a natural (Hausdorff) topology from that of G. Namely, if {γ_{1}, …, γ_{n}} is a set of generators of π then the evaluation map

\begin{matrix}Hom(π, G) & ⟶ G^{n} \\ ρ & ⟼ (ρ(γ_{1}), …, ρ(γ_{n})) \end{matrix}

is an embedding which induces a topology on Hom(π, G) that is independent on the choice of generators. Furthermore, if G is an algebraic group, then there is an induced algebraic structure on Hom(π, G), and we refer to the resulting object as the variety of representations.

However, the topology of the space Hom(π, G)/G inherited from Hom(π, G) could be rather pathological (e.g. may not be even Hausdorff). When G is a reductive linear algebraic group, i.e a subgroup of GL(n, ℝ), however, one can consider the algebraicgeometric quotient Hom(π, G)//G, which is defined as the variety whose coordinate ring is precisely the ring of G-invariant functions on Hom(π, G). Equivalently, it is the space of semi-stable orbits. In this case, the orbit of a representation ρ is semi-stable if ρ is a completely reducible representation. We call Hom(π, G)//G the character variety of π.

2.4. Obstruction Classes and Components of Hom(π, G). Let S be a connected surface and let π denote its fundamental group. Let G be a real algebraic Lie group. An element ρ in the space Hom(π, G) determines a flat prinipal G-bundle over S. This bundle gives rise to the obstruction classes in H^{q}(S, π_{q−1}(G)) and thus induces an obstruction class map

o_{q} : Hom(π, G) ⟶ H^{q}(S, π_{q−1}(G))

In particular, let G_{0} be the identity component for G and G/G_{0} = π_{0}(G) be the group of components of G. Then o_{1}(ρ) ∈ H^{1}(S, π_{0}(G)), which via the Hurewicz isomorphism is Hom(π, G/G_{0}). Thus, o_{1}(ρ) is just the composite

π ⟶^{ρ} G ⟶ G/G_{0}.

The map o_{1} is continuous, hence, constant on each connected component. For example if G = PGL(2, ℝ) ≃ SO(2, 1), then π_{0}(G) = ℤ_{2} and o_{1}(ρ) ∈ H^{1}(S, ℤ_{2}). In that case the action of PGL(2, ℝ) on hyperbolic 2-space H^{2} gives rise to an associated circle bundle (with fiber ∂H^{2} = ℝℙ^{1}), whose first Stiefel-Whitney class corresponds to o_{1}(ρ) (see Steenrod [19], §38; Goldman [5], §2). If n > 0 then π is a free group of rank r = 1−χ(S), where χ(S) is the Euler characteristic of S. Thus H^{1}(S, ℤ_{2}) ≃ ℤ_{2}^{r}, which is a free ℤ_{2}- module. In this case Hom(π, G) ≅ G^{r} and the fibers of o_{1} identify with the connected components of Hom(π, G). In this context we shall refer to o_{1} as the first Stiefel-Whitney class map.

2.5. SL(2, ℂ)-character Varieties. The example in the last paragraph is of particular geometric interest. Consider a finitely generated group π and the representation space Hom(π, SL(2, ℂ)). Since SL(2, ℂ) acts as a group of symmetries of hyperbolic 3-space H^{3}, the orbit space Hom(π, SL(2, ℂ))/ SL(2, ℂ) contains a subset that parametrizes equivalence classes of hyperbolic structures on 3-manifolds with fundamental group isomorphic to π.

The character variety

𝒳 = Hom(π, SL(2, ℂ))// SL(2, ℂ)

admits an embedding of 𝒳 as an algebraic subset of affine space. In particular, traces of a finite generating set of π. define coordinates on (X) (Procesi [17]). We consider the case when π is free of rank 2:

Theorem 2.1 (Fricke-Klein). Let X, and Y be the generators of π and let G = SL(2, ℂ). Then the character map:

χ : Hom(π, G)//G ⟶ ℂ^{3}
[ρ] ⟼ [\begin{matrix}ξ(ρ) \\ η(ρ) \\ ζ(ρ) \end{matrix}] = [\begin{matrix}tr(ρ(X)) \\ tr(ρ(Y)) \\ tr(ρ(XY)) \end{matrix}]

is an isomorphism. (Compare the discussion in Goldman [5], 4.1, [6],§§4–5 and [7].) Thus the traces of X, Y, XY parametrize Hom(π, G)//G as the affine space ℂ^{3}. In particular, if w(X, Y) is any word in X and Y, then tr(ρ(w(X, Y))) is expressed as a polynomial f_{w} in ξ, η, ζ.

Let ℋ be a copy of the hyperbolic plane H^{2} sitting inside H^{3}. The stabilizer of ℋ in SL(2, ℂ) is isomorphic to PGL(2, ℝ) and in this way PGL(2, ℝ) identifies with the full group of isometries of H^{2}. Naturally, we are interested in the orbits of the action of PGL(2, ℝ) by conjugation on Hom(π, PGL(2, ℝ)). However, in order to use the trace parametrization of the orbit space provided by the theorem of Fricke-Klein, we need to work with the relevant representation space, which in this case is Hom(π, SL_{±}(2, ℝ)). The subgroup

SL_{±}(2, ℝ) := {A ∈ GL(2, ℝ) | det(A) = ±1}

of GL(2, ℝ) is a double cover of PGL(2, ℝ). Since π is a free group, any representation ρ ∈ Hom(π, PGL(2, ℝ)) lifts to a representation in Hom(π, SL_{±}(2, ℝ)). Thus a conjugacy class of representations in Hom(π, PGL(2, ℝ)) corresponds to a conjugacy class of representations in Hom(π, SL_{±}(2, ℝ)) together with a choice of a lift.

2.6. SL_{±}(2, ℝ)-character Varieties. Let SL_{−}(2, ℝ) denote the subset of GL(2, ℝ) consisting of matrices of determinant −1. The group SL_{±}(2, ℝ) is isomorphic to

ISL(2, ℝ) = SL(2, ℝ) ⨿ i SL_{−}(2, ℝ)

and in this context we identify the two as subgroups of G = SL(2, ℂ). In view of the discussion in section 2.4, the representation space

ℛ = Hom(π, SL_{±}(2, ℝ)) ≅ Hom(π, ISL(2, ℝ)) ≅ ISL(2, ℝ) × ISL(2, ℝ)

has four connected components indexed by the elements of ℤ_{2} × ℤ_{2}. Let G_{0} = SL(2, ℝ) and G_{1} = i SL_{−}(2, ℝ). Then the correspondence defined by the Stiefel-Whitney class map is

ℛ_{j,k} = G_{j} × G_{k} ⟼ (j, k) ∈ ℤ_{2} × ℤ_{2}

where j, k ∈ {0, 1}. The following Proposition shows that the the restriction of the character map χ to ℛ can be used to parametrize conjugacy classes of ISL(2, ℝ) representations (compare Xia [22], pp. 10-13).

Proposition 2.2. Let χ_{jk} be the restriction of χ to ℛ_{j,k}. Assume (j, k) ≠ (0, 0). Then (1) χ_{jk} is surjective

(2) The image of χ_{jk} is

𝒳_{j,k} := χ(ℛ_{j,k}) = \{ \begin{matrix}ℝ × iℝ × iℝ & if & (j, k) = (0, 1) \\ iℝ × ℝ × iℝ & if & (j, k) = (1, 0) \\ iℝ × iℝ × ℝ & if & (j, k) = (1, 1) \end{matrix}

(3) Let ρ ∈ ℛ. Then

(2.2)  κ ∘ χ(ρ) = \{ \begin{matrix}x^{2} − y^{2} − z^{2} + xyz − 2 & if & ρ ∈ ℛ_{0,1} \\ −x^{2} + y^{2} − z^{2} + xyz − 2 & if & ρ ∈ ℛ_{1,0} \\ −x^{2} − y^{2} + z^{2} + xyz − 2 & if & ρ ∈ ℛ_{1,1} \end{matrix}

(4) Let u ∈ 𝒳_{j,k} be such that k(u) ≠ 2. Then ISL(2, ℝ) acts transitively on χ_{jk}^{−1}(u).

Proof. We refer the proof to Xia [22], Proposition 12 and 13. □

By a result of Culler and Shalen, a representation in ρ ∈ ℛ is reducible if and only if κ(χ(ρ)) = 2. Thus Proposition 2.2 implies that if u ∈ 𝒳_{j,k} is not in κ^{−1}(2), then the fiber χ_{jk}^{−1}(u) is an ISL(2, ℝ)-conjugacy class of irreducible representations in ℛ_{j,k}. In that sense 𝒳_{j,k} identifies with a component of the ISL(2, ℝ)-character variety of π.

Let κ_{jk} be the restriction of κ to 𝒳_{j,k}. Since 𝒳_{j,k} is naturally isomorphic to ℝ^{3} we will in fact consider the fibers of κ_{jk} as subsets of ℝ^{3}. Furthermore, whenever it is clear from the context, we will drop the subscript and use κ to denote the relevant polynomial as prescribed in (2.2).

2.7. Surfaces Whose Fundamental Group is Free of Rank 2. Let S be a surface whose fundamental group π is free of rank 2. Then S is precisely one of the following: a punctured^{2} torus S_{1,1}, a pair of pants S_{0,3}, a punctured Klein bottle S_{1,1}^{♯}, or a punctured Möbius band S_{0,2}^{♯}. These are all connected surfaces with Euler characteristic −1.

Remark. The notation used for referencing these surfaces is based on the genus and the number of boundary components of each surface. Thus S_{g,n} (respectively S_{g,n}^{♯}) denotes an orientable (respectively non-orientable) surface of genus g and n boundary components. For non-orientable surfaces, however, the notion of genus is not very consistently used throughout the literature. Some authors define genus as the number of copies of the projective plane in the cross-cap decomposition of the surface. In that context, the genus of the Klein bottle would be 2, and the genus of the Möbius band would be 1.

2In this context a “punctured surface” shall mean a surface with a topological disk removed

We adhere to the definition that is consistent with the formula for the Euler characteristic of a non-orientable surface S

χ(S) = 1 − β_{1}(S) − n

where β_{1}(S) is the first Betti number and n is the number of boundary components. We take β_{1}(S) as a definition for the genus of a non-orientable surface S (see Moise [14], § 22).

Next, we show how the geometry of each surface is related to the presentation of π.

Definition. An E-piece is a right-angled hyperbolic hexagon whose boundary is a piecewisegeodesic closed curve without self-intersections. Each edge of an E-piece is given the orientation of the underlying geodesic segment.

Each surface of Euler characteristic −1 can be realized by pasting two E-pieces subject to a certain gluing scheme. Figure 1 illustrates this idea for the non-orientable cases. Let

c′1 c′2 c′3  a′3  c{′′ \atop 1} c{′′ \atop 2} a{′′ \atop 3} a{′′ \atop 1} a′1  c{′′ \atop 3} β′′ β′ x0  γ′′ γ′  a′2  a{′′ \atop 2}

(a) Punctured Klein-bottle

c′2 c′3  a′3  c{′′ \atop 1} c{′′ \atop 2} a{′′ \atop 3} a′1  c{′′ \atop 3} β′′ x0  γ′′ a′2  a{′′ \atop 2} β′ γ′  β′ γ′ c′1  a{′′ \atop 1}

(b) Punctured Möbius Band

Figure 1: Constructing non-orientable surfaces from E-pieces

ρ ∈ Hom(π, PGL(2, ℝ)) be a discrete S-embedding. The union of two E-pieces E′ and E′′ is a fundamental region for ρ(π) acting on H^{2} with quotient S. From the gluing diagrams we obtain a geometric presentation of π in each case of a surface of Euler characteristic −1. For example, suppose S = S_{1,1}^{♯} is the punctured Klein-bottle. Then, S = E′ + E′′ modulo the gluing pattern shown on Fig. 1(a). In the quotient, β′∗β′′ = β, and γ′∗γ′′ = γ become the two generators of π, and the boundary geodesic that represents the puncture is δ = c′_{2} ∗ c′′_{3} ∗ c′_{1} ∗ c′′_{1} ∗ c′_{3} ∗ c′′_{2}. Here the “∗” operator indicates the obvious left-to-right curve concatenation with respect to a suitable parametrization on each geodesic segment. Thus, for a fixed base point x_{0}, we obtain a presentation of π ≡ π_{1}(S, x_{0}) as follows:

(2.3)  π = ⟨β, γ, δ | γβ^{2}γ = δ⟩

Similarly, from the diagram in Fig. 1(b) we obtain the following presentation of the fundamental group π ≡ π_{1}(S_{0,2}^{♯}, x_{0}) of the punctured Möbius band:

(2.4)  π = ⟨β, γ, δ_{1}, δ_{2} | δ_{1} = βγ, δ_{2} = βγ^{−1}⟩

where again β = β′ ∗ β′′ and γ = γ′ ∗ γ′′ are the two generators of π, while δ_{1} = c′_{3} ∗ c′′_{3} and δ_{2} = c′_{2} ∗ c′′_{2} ∗ c′_{1} ∗ c′′_{1} represent the two boundary components of S_{0,2}^{♯}.

2.8. Mapping Class Group, and the Structure of Out(π). For a compact connected surface S, the Mapping Class Group of S is defined as the group π_{0}(Homeo(S, ∂S)) of isotopy classes of homeomorphisms on S. There is a well-defined homomorphism:

(2.5)  N : π_{0}(Homeo(M)) ⟶ Out(π) ≅ Aut(π)/ Inn(π).

If S is a closed surface then by Dehn (unpublished) and Nielsen [16], N is an isomorphism. When ∂S ≠ ∅, then each component ∂_{i}S determines a conjugacy class C_{i} of elements of π_{1}(S) and the image of N consists of elements of Out(π) represented by automorphisms which preserve each C_{i}. Another theorem of Nielsen [15], implies that when S is a punctured torus, N is also an isomorphism.

The action of Out(π) on homology H_{1}(S; ℤ) ≅ ℤ^{2} defines a homomorphism

(2.6)  h : Out(π) ⟶ GL(2, ℤ).

which, again by Nielsen [15], is an isomorphism (see also Magnus-Karrass-Solitar [13], § 3.5, Corollary N4). Thus for a free group π of rank 2, Out(π) is isomorphic to GL(2, ℤ).

2.9. The modular group. In general the group Aut(π) acts on the SL(2, ℂ) character variety 𝒳 by:

ϕ_{∗}([ρ]) = [ρ ∘ ϕ^{−1}]

Let γ ∈ π and let ι_{γ} denote conjugation by γ. Since

ρ ∘ ι_{γ} = ι_{ρ(γ)} ∘ ρ

the subgroup Inn(π) acts trivially. Thus Out(π) acts on ℂ^{3} and since an automorphism ϕ of π is determined by

(ϕ(X), ϕ(Y)) = (w_{1}(X, Y), w_{2}(X, Y)),

for some words w_{1}, w_{2} in the generators, the action of ϕ on ℂ^{3} is given by a triple of polynomials

(f_{w_{1}}(ξ, η, ζ), f_{w_{2}}(ξ, η, ζ), f_{w_{1}w_{2}}(ξ, η, ζ))

Hence Out(π) acts on ℂ^{3} by polynomial automorphisms. Nielsen’s theorem (see Magnus-Karras-Solitar [13], Theorem 3.9) implies that any such automorphism preserves κ : ℂ^{3} ⟶ ℂ, that is

κ ((f_{w_{1}}(ξ, η, ζ), f_{w_{2}}(ξ, η, ζ), f_{w_{1}w_{2}}(ξ, η, ζ))) = κ(ξ, η, ζ).

Horowitz[9] determined the group Aut(ℂ^{3}, κ) of polynomial mappings ℂ^{3} ⟶ ℂ^{3} preserving κ. Let 𝔖_{3} be the symmetric group consisting of permutations of the coordinates ξ, η, ζ. Horowitz proved that the automorphism group of (ℂ^{3}, κ) is generated by the linear automorphism group

Aut(ℂ^{3}, κ) ∩ GL(3, ℂ) = (ℤ_{2} ⊕ ℤ_{2}) ⋊ 𝔖_{3}

and the quadratic reflection:

[\begin{matrix}ξ \\ η \\ ζ \end{matrix}] ⟶ [\begin{matrix}ξ \\ η \\ ξη − ζ \end{matrix}] ,

This group is commensurable with Out(π). The factor ℤ_{2}⊕ℤ_{2} corresponds to sign change automorphisms of ℛ. In particular, the three non-trivial elements (0, 1), (1, 0), (1, 1) act on representations by σ_{10}, σ_{01}, σ_{11} respectively:

σ_{01} · ρ : \begin{cases}X & ⟼ ρ(X) \\ Y & ⟼ −ρ(Y) \end{cases}

σ_{10} · ρ : \begin{cases}X & ⟼ −ρ(X) \\ Y & ⟼ ρ(Y) \end{cases}

σ_{11} · ρ : \begin{cases}X & ⟼ −ρ(X) \\ Y & ⟼ −ρ(Y) \end{cases}

The corresponding action on characters is:

(σ_{01})_{∗} : [\begin{matrix}ξ \\ η \\ ζ \end{matrix}] ⟼ [\begin{matrix}ξ \\ −η \\ −ζ \end{matrix}]

(σ_{10})_{∗} : [\begin{matrix}ξ \\ η \\ ζ \end{matrix}] ⟼ [\begin{matrix}−ξ \\ η \\ −ζ \end{matrix}]

(σ_{11})_{∗} : [\begin{matrix}ξ \\ η \\ ζ \end{matrix}] ⟼ [\begin{matrix}−ξ \\ −η \\ ζ \end{matrix}] .

Note that the sign change automorphisms are not induced by automorphisms of π. Thus the group Aut(ℂ^{3}, κ) is isomorphic to a semidirect product

PGL(2, ℤ) ⋉ (ℤ/2 ⊕ ℤ/2)

See Goldman [8], §1.3 and Appendix, for a detailed discussion.

The action of Aut(ℂ^{3}, κ) on ℂ^{3} restricts to an action on the ISL(2, ℝ)-character variety. However the subgroup 𝔖_{3} does not preserve the individual components 𝒳_{j,k}. For example, the automorphism induced by transposition of x and y maps characters in 𝒳_{1,0} to characters in 𝒳_{0,1}, and stabilizes 𝒳_{1,1}. For each component 𝒳_{j,k} there is a subgroup of finite index of Aut(ℂ^{3}, κ) that preserves 𝒳_{j,k}. We call this subgroup the modular group of 𝒳_{j,k} and denote it by Γ_{jk}.

For instance, Γ_{11} is generated by the quadratic reflections

Q_{x} : [\begin{matrix}x \\ y \\ z \end{matrix}] ⟶ [\begin{matrix}yz − x \\ y \\ z \end{matrix}] , Q_{y} : [\begin{matrix}x \\ y \\ z \end{matrix}] ⟶ [\begin{matrix}x \\ xz − y \\ z \end{matrix}] , Q_{z} : [\begin{matrix}x \\ y \\ z \end{matrix}] ⟶ [\begin{matrix}x \\ y \\ −xy − z \end{matrix}] ,

the sign-change automorphisms σ_{jk}, and the transposition

t_{xy} : [\begin{matrix}x \\ y \\ z \end{matrix}] ⟶ [\begin{matrix}y \\ x \\ z \end{matrix}]

where x, y, z ∈ ℝ. Since Γ_{01} ≅ Γ_{10} ≅ Γ_{11}, whenever it is clear from the context, we will drop the subscript and use Γ to denote the relevant modular group for the given component.

2.10. Goldman’s Result on the Real SL(2)-Characters of the Punctured Torus. Recently, Goldman [8] studied the action of the modular group on the SL(2, ℝ) component of 𝒳. In this case the modular group Γ is isomorphic to

PGL(2, ℤ) ⋉ (ℤ/2 ⊕ ℤ/2)

We summarize Goldman’s results in the theorem below.

Theorem 2.3 (Goldman). Let κ(ξ, η, ζ) = ξ^{2} + η^{2} + ζ^{2} − ξηζ − 2 and let c ∈ ℝ. ▷ For c < −2, the group Γ acts properly and freely on κ^{−1}(c) ∩ ℝ^{3}; ▷ For −2 ≤ c < 2, there is a compact connected component K_{c} of κ^{−1}(c) ∩ ℝ^{3} and

Γ acts properly and freely on the complement κ^{−1}(c) ∩ ℝ^{3} − K_{c}; ▷ For 18 ≥ c > 2, the group Γ acts ergodically on κ^{−1}(c) ∩ ℝ^{3}; ▷ For c > 18, the group Γ acts properly and freely on an open subset Ω_{c} ⊂ κ^{−1}(c)∩ℝ^{3}

and acts ergodically on the complement of Ω_{c}.

3. Fricke Spaces Within the Character Variety

We now focus on the components 𝒳_{j,k} of the ISL(2, ℝ)-character variety of π that correspond to non-zero Steifel-Whitney classes in ℤ_{2} × ℤ_{2}. Our goal is to identify the regions in 𝒳_{j,k} that parametrize the Fricke spaces of the two non-orientable surfaces whose fundamental group is isomorphic to π, namely S_{1,1}^{♯}and S_{0,2}^{♯}(see section 2.7). For short, we call these surfaces the PK-bottle and the PM-band respectively.

3.1. The Fricke Space of the Punctured Klein-Bottle. We first derive a presentation of the fundamental group of the PK-bottle S_{1,1}^{♯} from a presentation of π_{1}(S_{0,3}) via the Higman-Neumann-Neumann (HNN) extension construction. Let δ, α′, and α′′ be the homotopy classes in π_{1}(S_{0,3}) of the boundary loops of S_{0,3}. Then π_{1}(S_{0,3}) admits the following (redundant) presentation:

π_{1}(S_{0,3}) = ⟨δ, α′, α′′ | δα′α′′ = I⟩

The PK-bottle S_{1,1}^{♯} is obtained from the pair of pants S_{0,3} via identifying the boundary components α′ and α′′ by a (orientation reversing) diffeomorphism ϕ (see Fig 3.1). Let H = ⟨α′⟩ be the cyclic subgroup of π_{1}(S_{0,3}) generated by α′. Then the induced homomorphism ϕ_{∗} sends α′ to α′′ and thus its restriction to H is a monomorphism:

ϕ_{∗}|_{H} : H ⟼ π_{1}(S_{0,3})

The corresponding HNN-extension of π_{1}(S_{0,3}) is given by:

(3.1)  π_{1}(S_{0,3}) ∗_{ϕ} H = ⟨δ, α′, α′′, β | δα′α′′ = I, ϕ_{∗}(α′) := α′′ = βα′β^{−1}⟩

This defines a presentation of π_{1}(S_{1,1}^{♯}) which reduces to the more familiar:

(3.2)  π_{1}(S_{1,1}^{♯}) = ⟨α, β, δ | αβαβ^{−1} = δ^{−1}⟩

Both π_{1}(S_{0,3}) and π_{1}(S_{1,1}^{♯}) are free of rank two. In particular, π_{1}(S_{0,3}) is freely generated by α′ and α′′, while π_{1}(S_{1,1}^{♯}) is freely generated by α and β. The quotient map

π_{1}(S_{0,3}) ⟶ π_{1}(S_{0,3}) ∗_{ϕ} H

defines a monomorphism of the fundamental groups:

ϕ_{∗} : π_{1}(S_{0,3}) ⟶ π_{1}(S_{1,1}^{♯})
α′ ⟼ α
α′′ ⟼ βαβ^{−1}
δ ⟼ βα^{−1}β^{−1}α^{−1}

Let ρ ∈ ℛ_{0,1} and let X ∈ G_{0}, Y ∈ G_{1} denote the images under ρ of α and β respectively. Then the images under ρ ∘ ϕ_{∗} of α′ and α′′ will be X and Y XY^{−1} respectively. Thus the

α′  δ  α′′
Figure 2: S_{1,1}^{♯} as a quotient space of S_{0,3}

induced map ϕ^{∗} of representation spaces is:

ϕ^{∗} : ℛ_{0,1} ≅ G_{0} × G_{1} ⟶ G_{0} × G_{0} ≅ ℛ_{0,0}
(X, Y) ⟼ (X, Y XY^{−1})

A representation of π_{1}(S_{0,3}) defined by (X, Z) ∈ G_{0} × G_{0} ≅ ℛ_{0,0} pulls back to a representation of π_{1}(S_{1,1}^{♯}) in ℛ_{0,1} if and only if Z is in the centralizer of X in G_{1}, that is, there exists Y ∈ G_{1}, such that Z = Y XY^{−1}. The map ϕ^{∗} descends to a map (denoted again, by abuse of notation, as ϕ^{∗}) between the corresponding components of the character varieties:

ϕ^{∗} : 𝒳_{0,1} ≅ ℝ × iℝ × iℝ ⟶ ℝ^{3} ≅ 𝒳_{0,0}
[\begin{matrix}ξ \\ η \\ ζ \end{matrix}] ⟼ [\begin{matrix}ξ \\ ξ \\ −η^{2} − ζ^{2} + ξηζ + 2 \end{matrix}]

Setting ξ = x, η = iy, and ζ = iz with x, y, z ∈ ℝ establishes an isomorphism ℝ × iℝ × iℝ ≅ ℝ^{3} such that the map ϕ^{∗} can be expressed as follows:

ϕ^{∗} : 𝒳_{0,1} ≅ ℝ^{3} ⟶ ℝ^{3} ≅ 𝒳_{0,0}
[\begin{matrix}x \\ y \\ z \end{matrix}] ⟼ [\begin{matrix}x \\ x \\ y^{2} + z^{2} − xyz + 2 \end{matrix}]

A representation ρ ∈ ℛ_{0,0} gives rise to a hyperbolilc structure on int(S_{0,3}) if and only if ρ(α′), ρ(α′′) and ρ(δ) are hyperbolic with non-intersecting axes that span an ultra-ideal triangle in H^{2}. This condition is equivalent to χ([ρ]) ∈ (−∞, −2)^{3} ∪ (2, +∞)^{3}, such that odd number of components of χ([ρ]) are negative, or equivalently

(3.3)  |x| > 2
(3.4)  y^{2} + z^{2} − xyz + 2 < −2

But characters in 𝒳_{0,1} that lie in the fibers of ϕ^{∗} correspond to discrete embeddings of π_{1}(S_{1,1}^{♯}) in ℛ_{0,1} with quotient a PK-Bottle, and thus to marked hyperbolic structures on int(S_{1,1}^{♯}). Note that inequality 3.4 already implies that |x| > 2. We have thus proved the first part of the following

Proposition 3.1. Let x, y, z ∈ ℝ be the coordinate functions on 𝒳_{0,1}.

(1) The region Ω_{0}^{K} ⊂ ℝ^{3} parametrized by the inequality

y^{2} + z^{2} − xyz + 4 < 0

corresponds to discrete geometric K-embeddings inside ℛ_{0,1}.

(2) Let Γ_{K} be the image of the mapping class group Map(S_{1,1}^{♯})) in Γ under the Nielsen homomorphism and let Γ_{0}^{K} = Γ/Γ_{K} be the coset space of Γ_{K} in Γ. The space of all discrete K-embeddings identifies with

Ω^{K} = ∐_{γ∈Γ_{0}^{K}} γΩ_{0}^{K}

Proof. To prove part (2) we must show that if for some γ ∈ Γ the intersection Ω_{0}^{K} ∩ γΩ_{0}^{K} is nonempty, then γ ∈ Map(S_{1,1}^{♯}).Suppose that [ρ] ∈ Ω_{0}^{K} ∩ γΩ_{0}^{K}. The automorphism γ of the character space ℂ^{3} is induced by an automorphism γ̃ of π_{1}(S_{1,1}^{♯}) such that

[ρ_{0} ∘ γ̃] = γ ∘ [ρ]

for some ρ_{0} ∈ Ω_{0}^{K}. Thus ρ_{0} ∘ γ̃ is also a discrete geometric K-embedding with respect to the original peripheral structure. This means that γ̃ preserves the conjugacy class of each boundary component, and in particular that it represents an equivalence class in the mapping class group of S_{0,2}^{♯}. □

Once we have computed the Fricke space of the PK-bottle inside 𝒳_{0,1}, it is easy to obtain the Fricke spaces sitting in the other two non-zero components of the ISL(2, ℝ)- character variety. We show how to do this for 𝒳_{1,1}. The class of the automorphism

ψ : α ⟼ αβ^{−1}

of π induces the transposition

t_{xz} : [\begin{matrix}x \\ y \\ z \end{matrix}] ⟶ [\begin{matrix}z \\ y \\ x \end{matrix}]

on the character variety 𝒳. In particular t_{xz} interchanges 𝒳_{0,1} and 𝒳_{1,1} and consequently the image of Ω_{0}^{K} in 𝒳_{1,1} will be given by

(3.5)  x^{2} + y^{2} − xyz + 4 < 0

This corresponds to a change of marking, with respect to which the new presesntation of π_{1}(S_{1,1}^{♯}) becomes

⟨γ, β, δ | γβ^{2}γ = δ^{−1}⟩

3.2. The Fricke Space of the Punctured Möbius Band. We now determine the characters of the discrete geometric embeddings inside the (1, 1)-component of Hom(π, ISL(2, ℝ)). To this end, consider the inclusion of the fundamental group of the quadruply punctured sphere π_{1}(S_{0,4}) into π_{1}(S_{0,2}^{♯}) induced by the (double) covering map

q : S_{0,4} ⟶ S_{0,2}^{♯}

Suppose π_{1}(S_{0,2}^{♯}) is given the presentation discussed in section (2.7), namely:

(3.6)  π = ⟨β, γ, δ_{1}, δ_{2} | δ_{1} = βγ, δ_{2} = βγ^{−1}⟩

The orientable double cover of S_{0,2}^{♯}, which we denote for the moment by S̃_{0,2}^{♯}, is a connected surface of genus 0 with 4 boundary components and thus homeomorphic to S_{0,4}. The two sheets of the covering map, denoted L and L′, are represented schematically in Fig. 3.2. To obtain the covering surface, L and L′ are glued via identifying the two pairs of edges, respectively marked with ã′_{2} and ã′′_{2} (these are respectively the lifts of a′_{2} and a′′_{2} from Fig. 1(b)). After this identification, the geodesic segments d and d̂ concatenate to form a boundary loop, denoted by δ̃_{2}. Similarly, the geodesic segments d′ and d̂′ concatenate to form a boundary loop, denoted by δ̃′_{2}. Thus the boundary of S̃_{0,2}^{♯} consists of four components, namely δ̃_{1}, δ̃′_{1}, δ̃_{2}, and δ̃′_{2}. Their corresponding elements in the fundamental

group of S̃_{0,2}^{♯}, denoted again by δ̃_{1}, δ̃′_{1}, δ̃_{2}, and δ̃′_{2}, are in fact the lifts of δ_{1}, δ_{2} ∈ π_{1}(S_{0,2}^{♯}).

More precisely, fix a base point x_{0} ∈ S_{0,2}^{♯} and let x̃_{0} be the lift of x_{0} to L. Consider

the fundamental group π̃ of S̃_{0,2}^{♯} with base point x̃_{0}. Let x̃′_{0} be the lift of x_{0} to L′. In

γ̃ δ̃1  δ̃′1 d  ^{ã{′′ \atop 2}}  ^{ã′2} β̃  β̃ γ̃  d′ d̂ d̂′  ^{ã{′′ \atop 2}}^{ã′2}  ^{x̃′0} x̃0  L′ L

Figure 3: The Two Sheets of S_{0,4} as a Double Cover of S_{0,2}^{♯}

S̃_{0,2}^{♯} there are two homotopy classes of paths with initial point x̃_{0} and end point x̃′_{0}; one class projects to the generator β ∈ π, the other one projects to β^{−1}. Let β̃ and β̃′ be representatives of each class respectively. Similarly, let γ̃ and γ̃′ be representatives of the two homotopy classes of paths that project to γ and γ^{−1} respectively (see Fig. (3.2)).

With this notation, the homotopy classes of the boundary components of S̃_{0,2}^{♯} can be written as follows:

\begin{aligned}δ̃_{1} & = [β̃ ∗ (γ̃′)^{−1}] \\ δ̃_{2} & = [β̃ ∗ γ̃^{−1}] \end{aligned}  \begin{aligned}δ̃′_{1} & = [γ̃ ∗ (β̃′)^{−1}] \\ δ̃′_{2} & = [γ̃′ ∗ (β̃′)^{−1}] \end{aligned}

where ’∗’ denotes path concatenation. From these equations, the images of δ̃_{1}, δ̃′_{1}, δ̃_{2}, and δ̃′_{2} under the induced covering projection map q_{∗} can be easily computed:

\begin{matrix}q_{∗} : π_{1}(S̃_{0,2}^{♯}) & ⟶ π_{1}(S_{0,2}^{♯}) \\ [δ̃_{1} ; δ̃_{2} ; δ̃′_{1} ; δ̃′_{2}] & ⟼ [\begin{matrix}βγ \\ βγ^{−1} \\ γβ \\ γ^{−1}β \end{matrix}] \end{matrix}

But δ̃_{1}, δ̃′_{1}, δ̃_{2}, and δ̃′_{2} satisfy the relation

δ̃_{1}δ̃′_{2}(δ̃′_{1})^{−1}(δ̃_{2})^{−1} = I

and thus if we set

A = δ̃_{1}, B = δ̃′_{2}, C = (δ̃′_{1})^{−1}, D = (δ̃_{2})^{−1}

we obtain a presentation of π_{1}(S̃_{0,2}^{♯}) such that

π_{1}(S̃_{0,2}^{♯}) ≡ π_{1}(S_{0,4}) = ⟨A, B, C, D | ABCD = I⟩

The induced map q_{∗} is then uniquely determined by the images of the three generators A, B, and C

(3.7)  q_{∗} : [\begin{matrix}A \\ B \\ C \end{matrix}] ⟼ [\begin{matrix}βγ \\ γ^{−1}β \\ β^{−1}γ^{−1} \end{matrix}]

The image of q_{∗} in π_{1}(S_{0,2}^{♯}) is the index-2 subgroup consisting of words of even length,

and generated by βγ, γ^{−1}β, and β^{−1}γ^{−1}. The elements of π_{1}(S̃_{0,2}^{♯}) whose traces generate

the coordinate ring of the character variety of S̃_{0,2}^{♯}, are (see for instance Magnus [12])

A, B, C, D, AB, BC, CA

From (3.7) we compute the images under q_{∗} of the last four elements:

q_{∗} : [\begin{matrix}D \\ AB \\ BC \\ CA \end{matrix}] ⟼ [\begin{matrix}βγ^{−1} \\ β^{2} \\ γ^{−2} \\ β^{−1}γ^{−1}βγ \end{matrix}]

Thus the induced map q^{∗} on representation spaces:

q^{∗} : Hom(π_{1}(S_{0,2}^{♯}), G) ⟶ Hom(π_{1}(S_{0,4}), G)
ρ ⟼ ρ ∘ q_{∗}

can be expressed in terms of the images X = ρ(β) and Y = ρ(γ) of the generators β and γ as follows

q^{∗} : Hom(π_{1}(S_{0,2}^{♯}), G) ⟶ Hom(π_{1}(S_{0,4}), G)

[\begin{matrix}X \\ Y \end{matrix}] ⟼ [\begin{matrix}XY \\ Y^{−1}X \\ X^{−1}Y^{−1} \\ XY^{−1} \\ X^{2} \\ Y^{−2} \\ X^{−1}Y^{−1}XY \end{matrix} ]

Consequently, the induced map on the SL(2, ℂ)-character varieties, denoted again by q^{∗}, is defined in terms of trace coordinates as follows:

(3.8)  \begin{matrix}q^{∗} : Hom(π_{1}(S_{0,2}^{♯}), G)//G & ⟶ Hom(π_{1}(S_{0,4}), G)//G \\ ℂ^{3} ∋ [\begin{matrix}ξ \\ η \\ ζ \end{matrix}] & ⟼ [\begin{matrix}ζ \\ ξη − ζ \\ ζ \\ ξη − ζ \\ ξ^{2} − 2 \\ η^{2} − 2 \\ κ(ξ, η, ζ) \end{matrix}] ∈ ℂ^{7} \end{matrix}

where

ξ = tr(X), η = tr(Y), ζ = tr(XY)

Recall that the (1, 1)-component 𝒳_{1,1} of the ISL(2, ℝ)-character variety of S̃_{0,2}^{♯} is isomorphic to iℝ × iℝ × ℝ, and therefore identifies with ℝ^{3} by setting

ξ = ix, η = iy, ζ = z

with x, y, z ∈ ℝ. The restriction of q^{∗} to 𝒳_{1,1} can be expressed in terms of the x, y, z coordinates in the following way:

(3.9)  \begin{aligned}q^{∗} : & 𝒳_{1,1} ≅ ℝ^{3} & ⟶ & ℝ^{7} \\ & [\begin{matrix}x \\ y \\ z \end{matrix}] & ⟼ & [\begin{matrix}z \\ −xy − z \\ z \\ −xy − z \\ −x^{2} − 2 \\ −y^{2} − 2 \\ κ(ix, iy, z) \end{matrix}] \end{aligned}

The coordinate ring of the SL(2, ℂ)-character variety of S_{0,4} is generated by the traces a = tr(A), b = tr(B), c = tr(C), d = tr(D)

t_{AB} = tr(AB), t_{BC} = tr(BC), t_{CA} = tr(CA)

subject to the relation (see for instance Magnus ([12])):

t_{AB}^{2} + t_{BC}^{2} + t_{CA}^{2} + t_{AB}t_{BC}t_{CA}
= (ab + cd)t_{AB} + (ad + bc)t_{BC} + (ac + bd)t_{CA}
− (a^{2} + b^{2} + c^{2} + d^{2} + abcd − 4).

Thus the map

𝒳_{S_{0,4}}^{ℂ} = Hom(π_{1}(S_{0,4}), SL(2, ℂ))// SL(2, ℂ) ⟶ ℂ^{7}

defined by (a, b, c, d, t_{AB}, t_{BC}, t_{CA}) embeds 𝒳 onto a hypersurface in ℂ^{7}. The set of real points of 𝒳_{S_{0,4}}^{ℂ} parametrizes equivalence classes of representations in SL(2, ℝ). The following Proposition (compare Keen [11], Theorem 2) provides necessary and sufficient conditions for

(a, b, c, d, t_{AB}, t_{BC}) ∈ ℝ^{6}

to represent an equivalence class of discrete geometric Q-embeddings of π_{1}(S_{0,4}) into SL(2, ℝ).

Proposition 3.2. Given a < −2, b < −2, c < −2, d < −2, t_{AB} < −2, and t_{BC} < −2, there exist elements A^{∗}, B^{∗}, C^{∗}, D^{∗} ∈ SL(2, ℝ) such that

a = tr(A^{∗}), b = tr(B^{∗}), c = tr(C^{∗}), d = tr(D^{∗})
t_{A^{∗}B^{∗}} = tr(A^{∗}B^{∗}), t_{B^{∗}C^{∗}} = tr(B^{∗}C^{∗})

and such that F = ⟨A^{∗}, B^{∗}⟩ and F′ = ⟨C^{∗}, D^{∗}⟩ are Fuchsian groups of signature (0; 3). Moreover, the group H = F ∗_{ℤ} F′, which is the amalgamated product of F and F′ over the cyclic subgroups

ℤ ≅ ⟨A^{∗}B^{∗}⟩ ⊂ F, ℤ ≅ ⟨C^{∗}D^{∗}⟩ ⊂ F′,

is Fuchsian and represents a marked surface of signature (0; 4). Every such marked surface is so representable.

But a Fuchsian group H, whose quotient H^{2}/H is a marked hyperbolic surface S, gives rise to a discrete geometric embedding ρ ∈ Hom(π_{1}(S), SL(2, ℝ)) determined (up to conjugation) by the correspondence between the generators of π_{1}(S_{0,4}) and those of H. Thus we have the following

Corollary 3.3. Given a, b, c, d, t_{AB}, t_{BC} ∈ ℝ, such that

(3.10)  a < −2, b < −2, c < −2, d < −2, t_{AB} < −2, t_{BC} < −2,

there exists a hypebolic surface S_{0,4} of signature (0; 4) with marking Λ = {A, B, C, D} and a discrete Q-geometric embedding ρ ∈ Hom(π, SL(2, ℝ)), such that

a = tr(ρ(A)), b = tr(ρ(B)), c = tr(ρ(C)), d = tr(ρ(D))
t_{AB} = tr(ρ(AB)), t_{BC} = tr(ρ(BC))

All such discrete Q-geometric embeddings are obtained this way.

It follows from Corollary 3.3 that a real point on the character variety 𝒳_{S_{0,4}}^{ℂ}, that is,. a point p = (a, b, c, d, t_{AB}, t_{BC}, t_{CA}) ∈ ℝ^{7} whose coordinates satisfy the Fricke relation, corresponds to a discrete geometric Q-embedding if and only if its coordinates satisfy inequalities (3.10).

This conclusion together with the observation that some of the expressions that define the image of q^{∗} in (3.9) satisfy the given inequalities trivially, prove the first part of the following

Proposition 3.4. Let x, y, z ∈ ℝ be the coordinate functions on 𝒳_{1,1}.

(1) The region Ω_{0}^{M} of ℝ^{3} parametrized by the inequalities:

xy + z > 2
z < −2

corresponds to discrete geometric M-embeddings inside ℛ_{1,1}.

(2) Let Γ_{M} be the image of the mapping class group Map(S_{0,2}^{♯})) in Γ under the Nielsen homomorphism and let Γ_{0}^{M} = Γ/Γ_{M} be the coset space of Γ_{M} in Γ. The space of all discrete M-embeddings identifies with

Ω^{M} = ∐_{γ∈Γ_{0}^{M}} γΩ_{0}^{M}

The proof of the second part of the Proposition is analogous to the proof of Proposition 3.1

24 WILLIAM GOLDMAN AND GEORGE STANTCHEV

4. Classification of Characters and the τ-reduction Algorithm The next two sections discuss the Γ-action on the components

𝒳_{0,1} ∪ 𝒳_{1,0} ∪ 𝒳_{1,1}

of the ISL(2, ℝ)-character variety of π. We shall mainly work in 𝒳_{1,1}, but the results carry over verbatim to 𝒳_{0,1} and 𝒳_{1,0}.

Denote by “∼” the equivalence relation induced by the Γ-action on characters: For u, v ∈ 𝒳_{1,1}

u ∼ v ⟺ ∃γ ∈ Γ such that γu = v

Since κ is Γ-invariant, u ∼ v implies κ(u) = κ(v). In this section we show that when c < −14 there exist essentially two types of equivalence classes of characters upon each of which Γ acts in a substantially different manner.

Theorem 4.1. Suppose that u ∈ 𝒳_{1,1} satisfies κ(u) < −14. Then ∃u′ = (x′, y′, z′) ∼ u such that either:

M: u′ ∈ Ω_{0}^{M}, in which case u′ (and therefore u) is a character of a Fuchsian representation whose quotient is homeomorphic to a a once-punctured Möbius band; or

E: z̄′ ∈ (−2, 2) in which case u′ is the character of a representation mapping the

peripheral element βγ^{−1} ∈ π to an elliptic element of SL(2, ℝ). 4.1. Notation. For any (x, y, z) ∈ ℝ^{3}, let

z̄:= −xy − z

Since the quadratic reflection

Q_{z} : [\begin{matrix}x \\ y \\ z \end{matrix}] ⟶ [\begin{matrix}x \\ y \\ −xy − z \end{matrix}]

preserves every κ^{−1}(c) for c ∈ ℝ, it interchanges z and z̄. Fix x, y ∈ ℝ. Then z and z̄ are the two (necessarily real) roots of the quadratic polynomial

(4.1)  z^{2} + (xy)z − x^{2} − y^{2} − 2 − c

since

(4.2)  \begin{aligned}z + z̄ & = −xy \\ zz̄ & = −x^{2} − y^{2} − 2 − c. \end{aligned}

Thus Q_{z} is the deck transformation of the double covering Π : κ^{−1}(c) ⟶ Π(κ^{−1}(c)) obtained by restriction of the projection Π(x, y, z) = (x, y) to the xy-plane.

Denote by ℰ_{c} the region

{(x, y, z) ∈ κ^{−1}(c) | z̄ ∈ (−2, 2)}

4.2. The level sets of κ and the Fricke space of S_{0,2}^{♯}. Next we find a necessary condition for Ω_{0}^{M}and κ^{−1}(c) to have a non-empty intersection.

Lemma 4.2. Suppose u = (x, y, z) ∈ Ω_{0}^{M} ∩ κ^{−1}(c). Then c < −14

Proof. Since u ∈ Ω_{0}^{M}, by defintion z < −2, and z̄ < −2. Therefore z + z̄ < −4 and

zz̄ > 4. Then (4.2) imply that

(4.3)  \begin{aligned}xy & > 4 \\ x^{2} + y^{2} & < −c − 6 \end{aligned}

Since any x, y ∈ ℝ satisfy x^{2} + y^{2} ≥ 2xy,

c = −6 − (−c − 6) ≤ −6 − (x^{2} + y^{2}) ≤ −6 − 2xy < −14.
□

Corollary 4.3. Ω_{0}^{M} ∩ κ^{−1}(c) ≠ ∅ if and only if c < −14.

Remark. The sign change transformations

(4.4)  σ_{xz} : [\begin{matrix}x \\ y \\ z \end{matrix}] ⟼ [\begin{matrix}−x \\ y \\ −z \end{matrix}] , σ_{yz} : [\begin{matrix}x \\ y \\ z \end{matrix}] ⟼ [\begin{matrix}x \\ −y \\ −z \end{matrix}]

preserve κ and commute with Q_{z}. Since

z̄(σ_{xz}(u)) = z̄(σ_{yz}(u)) = −z̄(u)

it suffices to consider u ∈ 𝒳_{1,1} such that κ(u) < −14, z < −2 and z̄ > 2.

The proof of Theorem 4.1 proceeds in two steps: first we construct a function τ that is non-decreasing along certain subsets of the orbit of each character in Ω_{0}^{M}. Then, (always assuming c < −14 and u ∉ Ω_{0}^{M} ∪ ℰ_{c}), we find a finite sequence of characters u_{0} = u, u_{1}, … u_{N}, and a constant T > 0 such that τ(u_{i}) > τ(u_{i+2}) + T, and either u_{N} ∈ Ω_{0}^{M}, or u_{N} ∈ ℰ_{c}.

4.3. The Quadratic Reflections. Recall that, apart from Q_{z}, two other quadratic reflections act on characters as automorphisms of κ:

Q_{x} : [\begin{matrix}x \\ y \\ z \end{matrix}] ⟶ [\begin{matrix}yz − x \\ y \\ z \end{matrix}] and Q_{y} : [\begin{matrix}x \\ y \\ z \end{matrix}] ⟶ [\begin{matrix}x \\ xz − y \\ z \end{matrix}]

They are induced, respectively by the following automorphisms of π (cf. Goldman [8])

q_{x} : \begin{matrix}X & ⟼ & (XY)Y^{2}X(Y^{−1}X^{−1}) \\ Y & ⟼ & (XY X^{−1})Y^{−1}(XY^{−1}X^{−1}) \\ XY & ⟼ & (XY^{2})(XY)(Y^{−2}X^{−1}) \end{matrix}

and

q_{y} : \begin{matrix}X & ⟼ & (XY)X(Y^{−1}X^{−1}) \\ Y & ⟼ & (XY)X^{−1}Y^{−1}X^{−1}(Y^{−1}X^{−1}) \\ XY & ⟼ & (XY)^{−1} \end{matrix}

On the other hand Q_{z} is induced by the automorphism

q_{z} : \begin{matrix}X & ⟼ & X \\ Y & ⟼ & Y^{−1} \\ XY & ⟼ & XY^{−1} \end{matrix}

which interchanges the two boundary components of S_{0,2}^{♯}. Therefore its coset in Out(π) lies in the image of the mapping class group under the Nielsen homomorphism (2.5). In contrast q_{x} and q_{y} do not represent elements of the mapping class group of S_{0,2}^{♯}.

Clearly Q_{z} preserves Q_{x} and Q_{y} preserves Ω_{0}^{M}. Indeed, for each u ∈ Ω_{0}^{M}

z̄(Q_{x}(u)) = −y(yz − x) − z = −z(y^{2} + 1) + xy > 6

since (4.3) implies xy > 4 and z < −2. Similarly,

z̄(Q_{y}(u)) > 6

and therefore Q_{x}(u), Q_{y}(u) ∉ Ω_{0}^{M} ∪ ℰ_{c}.

4.4. The orbit as a binary tree. Consider the subgroup Λ ⊂ Γ generated by Q_{x}, Q_{y}, Q_{z}. Since Λ ≅ ℤ_{2} ∗ ℤ_{2} ∗ ℤ_{2}, it can be naturally associated with a trivalent tree, T_{Λ}(V, E), defined as follows. Each node v ∈ V represents a group element, that is, a reduced word on the generators. Nodes corresponding to words w_{1} and w_{2} respectively, are linked by an edge e ∈ E if there is a generator λ such that w_{1} = λw_{2}.

For any u ∈ Ω_{0}^{M}, the tree T_{Λ}(V, E) imparts a binary forest structure on the orbit Λ · u

Definition. Let u ∈ Ω_{0}^{M}. Define a binary tree B_{Λ}(u) inductively as follows

▷ u is the root of B_{Λ}(u)

▷ Q_{x}(u) and Q_{y}(u) are, respectively, the left and the right descendent of u.

ACTION OF THE MODULAR GROUP 27

▷ Suppose v is an arbitrary node and let v̂ be its parent. Let λ ∈ Λ be the generator such that v = λv̂. Then the descendents of v are λ_{1}v, and λ_{2}v, where λ_{1}, λ_{2} ≠ λ.

For every u ∈ Ω_{0}^{M} there is a “dual” tree B_{Λ}(Q_{z}(u)) rooted at Q_{z}(u). Since q_{x}, Q_{y}, Q_{z} freely generate Λ, the orbit Λ · u is the disjoint union of two binary trees

B_{Λ}(u) ⨿ B_{Λ}(Q_{z}(u))

Q_{y}  Q_{z} Q_{z}  Q_{x} Q_{y}  Q_{z}  Q_{z} Q_{y} Q_{x}  Q_{y}  Q_{x}  Q_{x} Q_{y}  Q_{z} Q_{z}  Q_{x} Q_{y}  Q_{z}  Q_{z} Q_{y} Q_{x}  Q_{y}  Q_{x}  Q_{x} Q_{z} Q_{x}  Q_{y}  Q_{x}  Q_{y} Q^{z}(u) u

Figure 4: The binary trees rooted at u and Q_{z}(u)

4.5. The τ-function.

Proposition 4.4. For every u ∈ Ω_{0}^{M}, the function

τ : 𝒳_{1,1} ⟶ ℝ
[\begin{matrix}x \\ y \\ z \end{matrix}] ⟼ −zz̄

does not decrease along the depth levels of B_{Λ}(u). More precisely, if v is a node of B_{Λ}(u), and v_{l}, v_{r} are its left and right descendents respectively, then

τ(v) ≤ τ(v_{l}), τ(v) ≤ τ(v_{r})

with at least one of the inequalities strict.

28 WILLIAM GOLDMAN AND GEORGE STANTCHEV

Proof. Fix u_{0} = (x_{0}, y_{0}, z_{0}) ∈ Ω_{0}^{M} and let u_{1} = (x_{1}, y_{1}, z_{1}) = Q_{x}(u_{0}). By definition z_{0} < −2 and z̄ < −2 and therefore

τ(u_{0}) = −zz̄ < 0

Then, x_{0}y_{0} = −(z_{0} + z̄_{0}) implies

(4.5)  z̄_{1} = −y_{0}^{2}z_{0} + x_{0}y_{0} − z_{0} = −z_{0}(y^{2} + 2) − z̄_{0} > −2z_{0} − z̄_{0} > 0

and consequently:

τ(u_{1}) > 0 > τ(u_{0})

Similar estimates apply verbatim in the case when u_{1} = Q_{y}(u_{0}) and therefore τ is strictly increasing at the root of B_{Λ}(u_{0}). However, the same type of algebraic argument does not extend to other nodes of B_{Λ}(u_{0}) directly. In order to proceed with the induction step we analyse the orbits of Λ geometrically.

Let L_{x−x_{0}}, L_{y−y_{0}}, L_{z−z_{0}} denote the level sets of the x-, y-, and z-coordinate functions respectively. For example

L_{x−x_{0}} = {(x, y, z) ∈ ℝ^{3} | x − x_{0} = 0}

Consider the conic h_{c,z_{0}} = L_{z−z_{0}} ∩ κ^{−1}(c) and its projection Π(h_{c,z_{0}}) to the xy-plane:

Π(h_{c,z_{0}}) = {(x, y) ∈ ℝ^{2} | k(x, y, z_{0}) = c}

Lemma 4.5. Let c < 2 and |z| > 2. Then h_{c,z} is a hyperbola with principal axes

(4.6)  a_{1} := {(x, y) | x + y = 0}, a_{2} := {(x, y) | x − y = 0}

and asymptotes

(4.7)  l_{1} := \{(x, y) | y = \frac{z − \sqrt{z^{2} − 4}}{2}x\}
l_{2} := \{(x, y) | y = \frac{z + \sqrt{z^{2} − 4}}{2}x\}.

When z < −2, the asymptotes l_{1} and l_{2} lie entirely in quadrant II and IV, and

h_{c,z} ∩ a_{1} = ∅, h_{c,z} ∩ a_{2} ≠ ∅

When z > 2, the asymptotes l_{1} and l_{2} lie entirely in quadrant I and III, and h_{c,z} ∩ a_{1} ≠ ∅, h_{c,z} ∩ a_{2} = ∅

a_{1}  l_{1} a_{2} l_{2}

(a) Level set z = −3

l_{2} l_{1}  a_{2} a_{1}

(b) Level set z = 3

Figure 5: Intersections of κ^{−1}(c) with z-coordinate level sets

Proof. Assume c < 2 and fix z_{0} such that |z_{0}| > 2. The quadratic form

S_{z_{0}}(x, y) = −x^{2} − y^{2} + z_{0}xy

has discriminant D_{z_{0}} = 1−z_{0}^{2}/4, which is negative for |z_{0}| > 2. Therefore S_{z_{0}} is indefinite and its level sets must be hyperbolae. The eigenvalues of S_{z_{0}} are

s_{1} = −\frac{z_{0}}{2} − 1 and s_{2} = \frac{z_{0}}{2} − 1

with eigenvectors

e_{1} = (\frac{−1}{\sqrt{2}} ; \frac{1}{\sqrt{2}} ) e_{2} = (\frac{1}{\sqrt{2}} ; \frac{1}{\sqrt{2}} )

Therefore with respect to the orthonormal basis (e_{1}, e_{2}), the form S_{z_{0}} has the following presentation

S_{z_{0}}(x̃, ỹ) = −(\frac{z_{0}}{2} + 1)x̃^{2} + (\frac{z_{0}}{2} − 1)ỹ^{2}

The principal axes a_{1} and a_{2} of h_{c,z} are, by definition, collinear with e_{1} and e_{2} respectively, which implies the first part of Lemma 4.5

30 WILLIAM GOLDMAN AND GEORGE STANTCHEV

Since by assumption z_{0}^{2} − c − 2 > 0, the equation of h_{c,z}

S_{z_{0}}(x̃, ỹ) + (z_{0}^{2} − c − 2) = 0

in terms of (e_{1}, e_{2}), has the following canonical form

(4.8)  \{ \begin{matrix}−\frac{x̃^{2}}{a_{c}^{2}(z_{0})} + \frac{ỹ^{2}}{b_{c}^{2}(z_{0})} & = 1, & if z_{0} < −2 \\ \frac{x̃^{2}}{a_{c}^{2}(z_{0})} − \frac{ỹ^{2}}{b_{c}^{2}(z_{0})} & = 1, & if z_{0} > 2 \end{matrix}

where

\begin{aligned}a_{c}(z_{0}) & = \sqrt{2 |\frac{(−z_{0}^{2} + c + 2)}{z_{0} + 2}|} \\ b_{c}(z_{0}) & = \sqrt{2 |\frac{(−z_{0}^{2} + c + 2)}{z_{0} − 2}|} \end{aligned}

Therefore in x̃, ỹ-coordinates, the asymptotes of h_{c}(z_{0}) are

l̃_{1,2} : a_{c}(z_{0})ỹ ∓ b_{c}(z_{0})x̃ = 0

from which equations (4.7) follow after applying the appropriate coordinate transformation. Similarly, the principal axes of h_{c}(z_{0}) are:

a_{1} : ỹ = 0, a_{2} : x̃ = 0

implying (4.6). The assertions about the intersections h_{c}(z_{0}) ∩ a_{1} and h_{c}(z_{0}) ∩ a_{2} follow from (4.8). □

Next consider the restriction of τ to h_{c,z}.

Definition. Let l be a line in ℝ^{2} with equation l(x, y) = 0. Then l partitions ℝ^{2} into half-planes

\begin{aligned}H_{l}^{+} & := \{(x, y) ∈ ℝ^{2} | l(x, y) > 0\} \\ H_{l}^{−} & := \{(x, y) ∈ ℝ^{2} | l(x, y) < 0\} \end{aligned}

the positive, and the negative half-plane associated with l respectively.

Lemma 4.6. Suppose c < 2, |z| > 2 and let h_{c,z} be as in Lemma 4.5. Let

a := \{ \begin{matrix}a_{1}, & if a_{1} ∩ h_{c,z} ≠ ∅ \\ a_{2}, & if a_{2} ∩ h_{c,z} ≠ ∅ \end{matrix}

and H_{a}^{+} and H_{a}^{−} as above. Then (1) The restriction τ|_{h_{c,z}} is invariant with respect to reflections in a_{1} and a_{2}.

(2) Let {P_{1}, P_{2}} = h_{c,z} ∩ a. Then τ|_{h_{c,z}} has global minima at P_{1}, P_{2}

(3) τ|_{h_{c,z}} decreases along h_{c,z} ∩ H_{a}^{−} and increases along h_{c,z} ∩ H_{a}^{+}.

Proof. With respect to the standard basis in ℝ^{2} reflection in a_{1} and a_{2} respectively are given by matrices

[\begin{matrix}−1 & 0 \\ 0 & −1 \end{matrix}] ,  [\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}] .

Each preserves z̄ = −xy − z as well as τ = −zz̄.

Next assume z < −2. Parametrize h_{c,z} as:

(4.9)  h_{c,z}(t) = [\begin{matrix}x(t) \\ y(t) \end{matrix}] = \frac{\sqrt{2}}{2} [\begin{matrix}−a_{c}(z) sinh t ± b_{c}(z) cosh t \\ a_{c}(z) sinh t ± b_{c}(z) cosh t \end{matrix}] , t ∈ ℝ.

For fixed z, the restriction τ|_{h_{c,z}} is a function of t alone:

τ|_{h_{c,z}}(t) = −zz̄ = x^{2} + y^{2} + c + 2 = b_{c}^{2}(z) cosh^{2} t + a_{c}^{2}(z) sinh^{2} t + c + 2.

The sign of the derivative of τ

\frac{dτ}{dt} = sinh 2t(a_{c}^{2}(z) + b_{c}^{2}(z))

depends only on the sign of sinh 2t. Thus

\frac{dτ}{dt} \{ \begin{matrix}< 0 & if and only if t < 0 \\ = 0 & if and only if t = 0 \\ > 0 & if and only if t > 0 \end{matrix}

Since

h_{c,z} ∩ H_{a}^{−} = {h_{c,z}(t) | t < 0}, h_{c,z} ∩ H_{a}^{+} = {h_{c,z}(t) | t > 0}

32 WILLIAM GOLDMAN AND GEORGE STANTCHEV

and {P_{1}, P_{2}} = h_{c,z}(0), the last two parts of Lemma 4.6 follow. A similar argument applies when z > 2. □

We complete the proof of Proposition 4.4. Assume c < 2 and |z_{0}| > 2. Let u_{0} = (x_{0}, y_{0}, z_{0}) lie on h_{c,z_{0}}. Since Q_{x} interchanges the two (necessarily real) roots of the quadratic polynomial

κ(x, y_{0}, z_{0}) − c = −x^{2} + (y_{0}z_{0})x − y_{0}^{2} + z_{0}^{2} − 2 − c

Q_{x} acts as a deck transformation of the double covering h_{c,z_{0}} ⟶ ℝ.. Thus l_{x} = L_{y−y_{0}} ∩ L_{z−z_{0}} intersects h_{c,z_{0}} precisely at two points: u_{0} and Q_{x}(u_{0}). Similarly l_{y} = L_{x−x_{0}} ∩ L_{z−z_{0}} intersects h_{c,z_{0}} at u_{0} and Q_{y}(u_{0}).

Claim. The points u_{0} and Q_{x}(u_{0}) lie on opposite branches of h_{c,z_{0}}. The points u_{0} and Q_{y}(u_{0}) lie on opposite branches of h_{c,z_{0}}.

The line at infinity l_{∞} in the projective completion ℝℙ^{2} of the affine plane L_{x−x_{0}} intersects h_{c,z_{0}} precisely at two points, the points of tangency of h_{c,z_{0}}with its asymptotes l_{1} and l_{2}. Denote these points by P_{1}^{∞} and P_{2}^{∞}. The point O = l_{1} ∩ l_{2} corresponds to the origin (0, 0, z_{0}) in L_{z−z_{0}}. The coordinate axes correspond to lines a_{x} and a_{y} respectively, both of which must intersect at O. Let P_{x}^{∞} = l_{∞} ∩ a_{x} and P_{y}^{∞} = l_{∞} ∩ a_{y}. Since in the affine plane h_{c,z_{0}} intersects both the x- and y-axis for any |z_{0}| > 2, it intersects a_{x} and a_{y} in ℝℙ^{2}. Therefore P_{x}^{∞} and P_{y}^{∞} lie in the interior of the segment P_{1}^{∞}P_{2}^{∞} (see Figure 6). l_{∞} partitions h_{c,z_{0}} into two disjoint arcs corresponding to the two branches of h_{c,z_{0}} in the affine plane. A line in ℝ^{2} intersects both branches of h_{c,z_{0}} if and only if the corresponding line in ℝℙ^{2} intersects l_{∞} in the interior of the segment P_{1}^{∞}P_{2}^{∞}. But u_{0} and Q_{x}(u_{0}) span a line parallel to the x-axis, so its corresponding line in ℝℙ^{2} intersects a_{x} at P_{x}^{∞}. Similarly, u_{0} and Q_{y}(u_{0}) span a line in ℝℙ^{2} intersecting a_{y} at P_{y}^{∞}. The proof of the claim is complete..

Therefore the action of

Λ_{x,y} = ⟨Q_{x}, Q_{y}⟩ ⊂ Λ

on the plane L_{z−z_{0}} ⊂ 𝒳_{1,1} reduces to a linear action of the infinite dihedral group on hyperbolae h_{c,z_{0}} (see Figure 7).

O u_{0} l_{∞}  l_{2} a^{y} a_{x}  P_{x}^{∞}  P_{y}^{∞}  P_{2}^{∞} l_{1}  P_{1}^{∞} Q_{x}(u_{0}) h_{c,z_{0}}  Q_{y}(u_{0})

Figure 6: Projective model of h_{c,z_{0}} and associated objects

Definition. We call the sets

\begin{aligned}𝒪_{x}(u) & = \{u, Q_{x}(u), Q_{y}Q_{x}(u) … \} \\ 𝒪_{y}(u) & = \{u, Q_{y}(u), Q_{x}Q_{y}(u) … \} \end{aligned}

respectively the x-forward and the y-forward Λ_{x,y}-orbit of u.

u_{0} u_{1}u_{2} u_{4} u_{3} h_{c,z}

Figure 7: Points on the Λ_{x,y}-orbit: u_{0}, u_{1} = Q_{x}(u_{0}), u_{2} = Q_{y}(u_{1}), etc

Lemma 4.6 implies that points on h_{c,z} may be partitioned into four types

(1) type (+,+): points u ∈ h_{c,z} such that

τ(Q_{x}(u)) − τ(u) > 0, τ(Q_{y}(u)) − τ(u) > 0

(2) type (+,-): points u ∈ h_{c,z} such that

τ(Q_{x}(u)) − τ(u) > 0, τ(Q_{y}(u)) − τ(u) < 0

(3) type (-,+): points u ∈ h_{c,z} such that

τ(Q_{x}(u)) − τ(u) < 0, τ(Q_{y}(u)) − τ(u) > 0

(4) type (0|0): points u ∈ h_{c,z} such that

either τ(Q_{x}(u)) − τ(u) = 0, or τ(Q_{y}(u)) − τ(u) = 0

Let 𝒯 denote the set {(+,+), (+,-), (-,+), (0|0)} of point types. Define a function θ : h_{c,z} ⟶ 𝒯 mapping each u ∈ h_{c,z} to the element of 𝒯 representing the type of u. We call θ the type assignment function on h_{c,z}.

Lemmas 4.5,4.6 provide a classification for the subsets of points of each type along h_{c,z}. Exactly four points have type (0|0): the x- and the y-intercepts of h_{c,z}. Let V_{l_{i},x} (respectively V_{l_{i},y}) denote the double cone spanned by the asymptote l_{i} and the x-axis

ACTION OF THE MODULAR GROUP 35

(respectively y-axis)^{3}. Assume z < −2. Then all points of type (+,+) lie in quadrant I and III. Points of type (+,-) and (-,+) lie inside V_{l_{1},y} and V_{l_{2},x} respectively. Assume z > 2. Then all points of type (+,+) lie in quadrant II and IV. Points of type (+,-) and (-,+) lie inside V_{l_{2},y} and V_{l_{1},x} respectively. Clearly, reflection in the principal axes a_{1} and a_{2} interchages the (+,-) and (-,+) types and leaves the (+,+) type invariant. A diagram representing points of different types is shown on Figure 8(a) for the case z < −2 and on Figure 8(b) for the case z > 2.

(−, +) (−, +)  (+, −) (+, −) (+, +)l_{1} (+, +)  l_{2}

(a) A z-level set when z < −2

(−, +) (+, +) (+, +) (+, −) (+, −) (−, +) l_{2} l_{1}

(b) A z-level set when z > 2

Figure 8: Classification of Points With Respect to the Monotonicity of τ

Lemma 4.2 implies that Ω_{0}^{M} lies in the cylinder

x^{2} + y^{2} < −c − 6, z ∈ ℝ

Therefore, if for some z < −2 and c < −14,

L_{z} ∩ Ω_{0}^{M} ∩ κ^{−1}(c) ≠ ∅,

then L_{z} ∩ Ω_{0}^{M} ∩ κ^{−1}(c) consists of two arcs of h_{c,z} inside the disk D_{c}^{Ω} : x^{2} + y^{2} < −c − 6.

3There are, naturally, two double cones associated with a pair of intersecting lines; we choose the one with the smaller cone angle

h_{c,z}  D_{c}^{Ω} h_{c,z} y_{2}^{D} y_{2}^{h}  x_{1}^{D} x_{1}^{h} y_{1}^{D} y_{1}^{h} x_{2}^{h} x_{2}^{D}
Figure 9: The set L_{z} ∩ Ω_{0}^{M} ∩ κ^{−1}(c) inside D_{c}^{Ω}

The x- and the y-intercepts of h_{c,z} are:

x_{1,2}^{h} = y_{1,2}^{h} = ±\sqrt{z^{2} − c − 2}

while the values of the x and the y-intercepts of ∂D_{c}^{Ω} are:

x_{1,2}^{D} = y_{1,2}^{D} = ±\sqrt{−c − 6}

Since \sqrt{z^{2} − c − 2} > \sqrt{−c − 6}, the arcs h_{c,z} ∩ D_{c}^{Ω} do not intersect the coordinate axes. Therefore all points in h_{c,z} ∩ D_{c}^{Ω} have type (+,+), and consequently

τ(u) < τ(Q_{x}(u)), τ(u) < τ(Q_{y}(u)), ∀u ∈ h_{c,z} ∩ D_{c}^{Ω}

Thus, when u ∈ Ω_{0}^{M} ∩ κ^{−1}(c), τ strictly increases at the initial point of each orbit 𝒪_{x}(u) and 𝒪_{y}(u). (Compare Proposition 4.4.) Furthermore suppose that u ∈ h_{c,z} satisfies

θ(Q_{x}(u)) ≠ (+,+) and θ(Q_{y}(u)) ≠ (+,+).

Then

\begin{aligned}θ(u) & = (-,+) ⇒ θ(Q_{x}(u)) = θ(Q_{y}(u)) = (+,-) \\ θ(u) & = (+,-) ⇒ θ(Q_{x}(u)) = θ(Q_{y}(u)) = (-,+) \end{aligned}

Therefore, when u ∈ Ω_{0}^{M} ∩ κ^{−1}(c) the image of 𝒪_{x}(u) under θ is the sequence

                       (+,+),(-,+),(+,-),(-,+), . . . .

ACTION OF THE MODULAR GROUP 37

The image of 𝒪_{y}(u) under θ is the sequence

                        (+,+),(+,-),(-,+),(+,-) . . .

Consequently, τ strictly increases along both 𝒪_{x}(u) and 𝒪_{y}(u).

We return to the the proof of Proposition 4.4. Let v be an arbitrary node of B_{Λ}(u). Let

P_{u,v} = {v_{0} = u, v_{1}, v_{2}, …, v_{N} = v}

be the (unique) path from u to v in B_{Λ}(u). Partition P_{u,v} into subsets

U_{0} = {v_{i_{0}} = v_{0}, …, v_{i_{1}}}, U_{1} = {v_{i_{1}+1}, …, v_{i_{2}}}, …, U_{m} = {v_{i_{m}+1}, …, v_{i_{m+1}} = v_{N}} such that v_{i_{j}+1} = Q_{z}(v_{i_{j}}) and U_{j} is a subset of either 𝒪_{x}(v_{i_{j}+1}), or 𝒪_{y}(v_{i_{j}+1}), for each j = 0, …, m. Since either U_{0} ⊂ 𝒪_{x}(u), or U_{0} ⊂ 𝒪_{y}(u), τ is strictly increasing along U_{0}. Furthermore, since τ is Q_{z}-invariant,

τ(v_{i_{j}}) = τ(v_{i_{j}+1})

for each j = 0, …, m. Also, by assumption z(v_{i_{1}}) = z(v_{0}) < −2, and z̄(v_{i_{1}}) = z̄(v_{0}) > 2. Consequently

z(v_{i_{1}+1}) = z(Q_{z}(v_{i_{1}})) = z̄(v_{i_{1}}) > 2

Recall that the hyperbola h_{c}(z_{i+1}) has the same principal axes as h_{c}(z_{i}), as prescribed by Lemma 4.5

\begin{aligned}a_{1} : x − y & = 0 \\ a_{2} : x + y & = 0 \end{aligned}

However the intersection properties of a_{j} with h_{c}(z_{i}) and h_{c}(z_{i+1}) are different, namely:

a_{1} ∩ h_{c}(z_{i}) ≠ ∅, a_{2} ∩ h_{c}(z_{i}) = ∅

while

a_{1} ∩ h_{c}(z_{i+1}) = ∅, a_{2} ∩ h_{c}(z_{i+1}) ≠ ∅

Thus all points of type (+,+) on h_{c}(v_{i_{1}+1}) lie in quadrant II and IV. Recall that v_{i_{1}} is a point of type either (+,-) or (-,+) and hence lies in quadrant II or IV. But then v_{i_{1}+1}, which is the image of v_{i_{1}} under Q_{z}, must also lie in quadrant II or IV and therefore it

38 WILLIAM GOLDMAN AND GEORGE STANTCHEV

must be of type (+,+). Hence τ must be strictly increasing along 𝒪_{x}(v_{i_{1}+1}) and 𝒪_{y}(v_{i_{1}+1}), and therefore along U_{1}. Similar argument applies to U_{j}, for 1 < j ≤ m. The proof of Proposition 4.4 is complete. □ 4.6. Growth of the τ function. So far we have shown that for each u ∈ Ω_{0}^{M}, the τ function is non-decreasing along the path P_{u,v} from u to an arbitrary node v in B_{Λ}(u). Moreover, τ is strictly increasing along P_{u,v} except for a (possibly empty) subset of nodes at which τ is constant. Next, we estimate the variation of τ among any pair of characters

(w_{0}, w_{1}) ∈ Δ_{c} := {(w_{0}, w_{1}) ∈ κ^{−1}(c) × κ^{−1}(c) | w_{1} = λw_{0}, λ ∈ {Q_{x}, Q_{y}}} such that |z(w_{0})| > 2.

Proposition 4.7. Suppose c < 2. Let w_{0} = (x_{0}, y_{0}, z_{0}) ∈ κ^{−1}(c) be such that |z_{0}| > 2. Then

(1) if w_{1} = (x_{1}, y_{0}, z_{0}) = Q_{x}(w_{0})

|τ(w_{1}) − τ(w_{0})| ≥ |y_{0}z_{0}| max (|y_{0}|\sqrt{z_{0}^{2} − 4}, 2\sqrt{z_{0}^{2} − c − 2})

(2) if w_{1} = (x_{0}, y_{1}, z_{0}) = Q_{y}(w_{0})

|τ(w_{1}) − τ(w_{0})| ≥ |x_{0}z_{0}| max (|x_{0}|\sqrt{z_{0}^{2} − 4}, 2\sqrt{z_{0}^{2} − c − 2})

Proof. Since |z_{0}| > 2 , the z_{0}-level set L_{z−z_{0}} ∩ κ^{−1}(c) is the hyperbola h_{c,z_{0}}. We prove case (2) first. Let A_{l_{1}} and A_{l_{2}} be the intersection points of the line l_{x} : x = x_{0} with l_{1} and l_{2} respectively. Clearly

|y_{1} − y_{0}| > d(A_{l_{1}}, A_{l_{2}}) = |\frac{z_{0} + \sqrt{z_{0}^{2} − 4}}{2}x_{0} − \frac{z_{0} − \sqrt{z_{0}^{2} − 4}}{2}x_{0}| = |x_{0}|\sqrt{z_{0}^{2} − 4}

where d(· , ·) denotes the (Euclidean) distance function in ℝ^{2}. On the other hand, since the minimum vertical distance between pairs of points on h_{c,z_{0}} is achieved at x_{0} = 0

|y_{1} − y_{0}| > min_{w∈h_{c,z_{0}}} |y(w) − y(Q_{y}(w))| = 2\sqrt{z_{0}^{2} − c − 2}

Therefore

(4.10)  |y_{1} − y_{0}| ≥ max (|x_{0}|\sqrt{z_{0}^{2} − 4}, 2\sqrt{z_{0}^{2} − c − 2})

By the defintion of τ

\begin{matrix}|τ(w_{1}) − τ(w_{0})| = & |z_{0}||(z̄_{1} − z̄_{0})| \\ = & |z_{0}||y_{0}||x_{1} − x_{0}| \end{matrix}

which together with inequality (4.10) implies part (2) of Proposition 4.7. Similar argu-

ment applies when w_{1} = Q_{x}(u), in which case

(4.11)  |x_{1} − x_{0}| ≥ max (|y_{0}|\sqrt{z_{0}^{2} − 4}, 2\sqrt{z_{0}^{2} − c − 2})

and

\begin{matrix}|τ(w_{1}) − τ(w_{0})| = & |z_{0}||(z̄_{1} − z̄_{0})| \\ = & |z_{0}||x_{0}||y_{1} − y_{0}| \end{matrix}

□

The assumption that |z| > 2 and c < 2 eliminates z from the estimates for the increment of τ.

Corollary 4.8. Let c < 2 be fixed, and let

(w_{0}, w_{1}) ∈ Δ_{c}

such that |z(w_{0})| > 2, and |z(w_{1})| > 2. Then the variation of τ is bounded from below by

a quantity depending on x_{0} or y_{0} alone. In particular

(4.12)  |τ(Q_{x}(w_{0})) − τ(w_{0})| ≥ 16|y_{0}|\sqrt{2 − c}

and

(4.13)  |τ(Q_{y}(w_{1})) − τ(w_{0})| ≥ 16|x_{0}|\sqrt{2 − c}

4.7. τ-reduction for Fricke-space characters. We have seen so far that the τ-function is non-decreasing along the levels of B_{Λ}(u) for any

u = (x, y, z) ∈ Ω_{0}^{M} ∩ κ^{−1}(c).

Also for every w ∈ B_{Λ}(u) ∩ Δ_{c}, and fixed c, the increment is bounded from below by a quantity depending on |x| or |y| alone. Since z < −2 and z̄ < −2,

zz̄ = −x^{2} − y^{2} − 2 − c > 4
z + z̄ = −xy < −4.

The first inequality implies that |x| < \sqrt{−c − 6} and |y| < \sqrt{−c − 6}, which together with the second inequality yield

\begin{aligned}|x| & > \frac{4}{\sqrt{−c − 6}} \\ |y| & > \frac{4}{\sqrt{−c − 6}}. \end{aligned}

Thus if κ^{−1}(c) ∩ Ω_{0}^{M} ≠ ∅ (or equivalently, if c < −14), there is a constant T_{c} such that

(4.14)  |τ(w_{1}) − τ(w_{0})| ≥ T_{c}

for every u ∈ Ω_{0}^{M} and a pair w_{0}, w_{1} of successive nodes in B_{Λ}(u) ∩ Δ_{c}. The uniform lower bound for the increment of τ along the depth levels of B_{Λ}(u) guarantees the following. The path P_{v,u} from an arbitrary character v in Λ · Ω_{0}^{M} to the unique character u ∈ Ω_{0}^{M} such that v = B_{Λ}(u), can be recovered in finite number of steps via reduction of τ by a definite amount at each step. More precisely,

Lemma 4.9. Let c < −14 and let v ∈ κ^{−1}(c). Assume v ∈ B_{Λ}(u) for some u ∈ Ω_{0}^{M}.

Then

P_{v,u} = {v_{0} = v, v_{1}, …, v_{n} = u}

where {v_{i}}_{i=0}^{n} is the (unique) sequence such that for each 0 ≤ i ≤ n − 1

v_{i+1} = \{ \begin{matrix}Q_{x}(v_{i}) & if τ(Q_{x}(v_{i})) < τ(v_{i}) \\ Q_{y}(v_{i}) & if τ(Q_{y}(v_{i})) < τ(v_{i}) \\ Q_{z}(v_{i}) & otherwise \end{matrix}

Moreover n depends only on c and v.

4.8. τ-reduction for arbitrary characters. This type of reduction process extends to arbitrary characters

v ∈ ℋ_{c} := κ^{−1}(c) ∩ (𝒳_{1,1} − Ω_{0}^{M} ∪ ℰ_{c} ∪ Q_{z}(ℰ_{c}))

By definition, v ∈ ℋ_{c} implies that |z(v)| > 2 and |z̄(v)| > 2. For such v, the proof of Proposition 4.4 produces a sequence

𝔪_{c}(v) : v_{0} = v, v_{1}, …, v_{n}, …

such that for each i = 1, 2, …:

▷ v_{i+1} = λ_{i}v_{i}, where λ_{i} ∈ {Q_{x}, Q_{y}, Q_{z}}

▷ λ_{i} ≠ λ_{i+1}
▷ τ(v_{i+1}) ≤ τ(v_{i})

Definition. We call such a sequence τ-minimizing. A τ-minimizing sequence is said to

terminate, if there exists n, such that one of the following occurs

▷ v_{n} ∈ ℰ_{c}, or
▷ v_{n} ∈ Ω_{0}^{M}, or
▷ v_{n} ∈ Σ_{0}^{c} := {(x, y, z) ∈ κ^{−1}(c) | x = 0, or y = 0}

The element v_{n_{t}} indexed by the smallest such n, will be called a terminator.

The sequence 𝔪_{c}(v) can be constructed as follows. Let v_{i} = (x_{i}, y_{i}, z_{i}), and assume |z_{i}| > 2 and |z̄_{i}| > 2. Then v_{i} lies on the hyperbola h_{c}(z_{i}). There are four possibilities based on the type of v_{i}

(1) v_{i} is of type (-,+); then v_{i+1} := Q_{x}(v_{i})

(2) v_{i} is of type (+,-); then v_{i+1} := Q_{y}(v_{i})

(3) v_{i} is of type (+,+); then v_{i+1} := Q_{z}(v_{i})

(4) v_{i} is of type (0|0); then v_{i} ∈ Σ_{0}^{c} and is therfore a terminator.

Proposition 4.10. After a finite number of steps, every τ-minimizing sequence termi-

nates.

Proof. Clearly, this is true if v ∈ Ω^{M} (Lemma 4.9). Thus assume that v ∉ Ω^{M}.

Recall that at each v_{i} where τ is strictly decreasing, Proposition 4.7 provides a lower

bound on |τ(v_{i}) − τ(v_{i+1})| that depends on x_{i} or y_{i} alone. As long as the x_{i}’s and the y_{i}’s

do not accumulate at 0, the decrement of τ by a definite amount at each step guarantees

that after finitely many steps there will be an element v_{n} such that τ(v_{n}) < 4 and hence

42 WILLIAM GOLDMAN AND GEORGE STANTCHEV

v_{n} will lie in one of Σ_{0}^{c}, ℰ_{c}, or Q_{z}(ℰ_{c}). Thus it suffices to show that if v ∉ Λ · Σ_{0}^{c}, then the x and the y coordinates of characters on the Λ-orbit of v but outside of ℰ_{c} are bounded away from 0 by some positive constant depending on v.

Lemma 4.11. For every v ∉ Ω^{M} ∪ ℰ_{c} ∪ Λ · Σ_{0}^{c} there exists positive constants ε_{x}(v) and ε_{y}(v) such that

inf_{λ∈Λ}{x(λv) | λv ∉ ℰ_{c}} = ε_{x}(v) > 0

inf_{λ∈Λ}{y(λv) | λv ∉ ℰ_{c}} = ε_{y}(v) > 0

Proof. We prove this by contradiction. Suppose no such constants exist. If |z| > 2 and |z̄| > 2, but h_{c,z} ∩ Ω_{0}^{M} = ∅, the set of points on h_{c,z} of type (+,+) partitions into three disjoint subsets

Z̄_{x} = {u ∈ h_{c,z} | Q_{z}(u) ∈ (−, +)}
Z̄_{y} = {u ∈ h_{c,z} | Q_{z}(u) ∈ (+, −)}
Z̄_{e} = {u ∈ h_{c,z} | −2 < z(Q_{z}(u)) < 2}

Definition. Points in Z̄_{x} (respectively Z̄_{y}, Z̄_{e}) will be called points of type z̄_{(−+)} (respectively z̄_{(+−)}, z̄_{(e)}). Notice that by definition Z̄_{e} ⊂ ℰ_{c}.

Fix c < 2 and consider the projection of κ^{−1}(c) onto the xy-plane

Π : κ^{−1}(c) ∋ (x, y, z) ⟼ (x, y) ∈ ℝ^{2}.

The level set L_{z+2} ∩ κ^{−1}(c) projects onto the pair of lines

h_{−2}^{±} : x + y ± \sqrt{−c + 2} = 0

and the set L_{z−2} ∩ κ^{−1}(c) projects onto the pair of lines

h_{+2}^{±} : x − y ± \sqrt{−c + 2} = 0.

The pair h_{−2}^{±} partitions the xy-plane into two regions:

H_{−2}^{−} = \{(x, y) ∈ ℝ^{2} | (x + y + \sqrt{−c + 2})(x + y − \sqrt{−c + 2}) < 0\}

ACTION OF THE MODULAR GROUP 43

and

H_{−2}^{+} = \{(x, y) ∈ ℝ^{2} | (x + y + \sqrt{−c + 2})(x + y − \sqrt{−c + 2}) > 0\}

The first one, H_{−2}^{−} is the “strip” bounded by h_{−2}^{±}, and the second one, H_{−2}^{+} is the

H_{−2}^{−}  H_{+2}^{−}  H_{+2}^{+} H_{+2}^{+} H_{−2}^{+} H_{−2}^{+} h_{+2}^{+} h_{+2}^{−}  h_{−2}^{+} h_{−2}^{−} P_{x}^{−} P_{y}^{−} P_{y}^{+} P_{x}^{+}

Figure 10: Lines, cones, and half-planes

complement of H_{−2}^{−} in ℝ^{2} (see Figure 10). Similarly, the pair h_{+2}^{±} partitions the plane into regions

H_{+2}^{−} = \{(x, y) ∈ ℝ^{2} | (x − y + \sqrt{−c + 2})(x − y − \sqrt{−c + 2}) < 0\}

and

H_{+2}^{+} = \{(x, y) ∈ ℝ^{2} | (x − y + \sqrt{−c + 2})(x − y − \sqrt{−c + 2}) > 0\}

When z_{0} < −2 (respectively z_{0} > 2), the level sets L_{z−z_{0}} ∩ κ^{−1}(c) = h_{c,z_{0}} project to a family of hyperbolae contained in H_{−2}^{+} (respectively H_{+2}^{+}). The regions H_{−2}^{−} and H_{+2}^{−} contain a family of ellipses, which are the projections of the level sets L_{z−z_{0}} ∩ κ^{−1}(c) for −2 < z_{0} < 2. Hence the set of z̄_{(e)} points projects onto the union

H_{−2}^{−} ∪ H_{+2}^{−}

If |z_{0}| > 2, the set of z̄_{(−+)} and z̄_{(+−)}points on h_{c,z_{0}} projects onto

H_{−2}^{+} ∩ H_{+2}^{+}

which is a union of four cones with vertices P_{x}^{±} = (±\sqrt{−c + 2}, 0) and P_{y}^{±} = (0, ±\sqrt{−c + 2}) (see Figure 10 and 11). In particular, the set of z̄_{(−+)} points projects onto the cones with

(a) for z < −2^{z̄_{(−+)} z̄_{(e)} z̄_{(−+)}  z̄_{(e)}  z̄_{(+−)} h_{−2}^{−} h_{−2}^{+} z̄_{(+−)}}  (b) for z > 2^{z̄_{(−+)}  z̄_{(e)} z̄_{(+−)}  z̄_{(−+)} z̄_{(e)}  z̄_{(+−)}}

Figure 11: Points of type z̄_{(−+)}, z̄_{(+−)}, and z̄_{(e)}

vertices P_{x}^{±}. The set of z̄_{(+−)} points projects onto the cones with vertices P_{y}^{±} (see Figure 12). Observe that the strip

F_{x} = ℝ × (−\sqrt{−c + 2}, \sqrt{−c + 2})

does not contain any projections of z̄_{(+−)} points. Similarly, the strip

F_{y} = (−\sqrt{−c + 2}, \sqrt{−c + 2}) × ℝ

does not contain any projections of z̄_{(−+)} points. Consequently, if v_{n_{1}} ∉ ℰ_{c} of the τminimizing sequence {v_{i}} enters the slab

F_{x} × ℝ

then it must remain on the same y-level set for all i ≥ n_{1}. Similarly, if an element v_{n_{2}} ∉ ℰ_{c} enters the slab

F_{y} × ℝ

\begin{aligned}h_{+2}^{+} & h_{+2}^{+} & h_{−2}^{−} & h_{−2}^{+} & P_{x}^{−} & P_{y}^{+} & P_{x}^{+} & P_{y}^{−} \\ & & & (a) for z < −2 \end{aligned}

\begin{aligned}h_{+2}^{+} & h_{+2}^{+} & h_{−2}^{−} & h_{−2}^{+} & P_{x}^{+} & P_{y}^{+} & P_{x}^{−} & P_{y}^{−} \\ & & & (b) for z > 2 \end{aligned}

Figure 12: Families of hyperbolae as projections of κ^{−1}(c) ∩ L_{z}

46 WILLIAM GOLDMAN AND GEORGE STANTCHEV

then it must remain on the same x-level set for all i ≥ n_{2}. Since we had assumed that either the x or the y-coordinates of characters in {v_{i}} accumulate at 0, there exists v_{n} ∉ ℰ_{c} ∪ Σ_{0}^{c}, n > 0, such that either

v_{n} ∈ F_{x} × ℝ

or

v_{n} ∈ F_{y} × ℝ

Suppose v_{n} ∈ F_{y} × ℝ. This means that x(v_{i}) must be constant and |y(v_{i})| > \sqrt{−c + 2} for all i > n, a contradiction. Similarly, v_{n} ∈ F_{x} × ℝ implies a contradiction. This concludes the proof of Lemma 4.11 and hence of Proposition 4.10. □

The proof of Lemma 4.11 suggests that if a τ-minimizing sequence enters one of the regions F_{x} × ℝ, or F_{y} × ℝ, its behavior can be determined explicitly. More precisely, let {v_{i}}, be a τ-minimizing sequence.

Definition. A terminal x-plane (respectively y-plane) is a level set L_{x−x_{0}} (respectively L_{y−y_{0}}) such that if v_{n} ∈ L_{x−x_{0}} (respectively v_{n} ∈ L_{y−y_{0}}) for some n, then v_{i} ∈ L_{x−x_{0}} (respecitvely v_{i} ∈ L_{y−y_{0}}), for all n ≤ i ≤ n_{t}, where n_{t} is the index of the terminating element of {v_{i}}.

Thus all elements of a τ-minimizing sequence that belong to

F_{x} × ℝ ∪ F_{y} × ℝ

must lie on a terminal plane. Figure 13 shows several elements of the minimizing sequence of the character u = (−0.2, 12, −10) on its terminal x-plane. Successive points are joined by line segments to facilitate visualization.

u = (−0.2, 12, −10)

Figure 13: The terminal plane of the character u = (−0.2, 12, −10)

4.9. The τ-Reduction Algorithm. We use the results obtained so far to construct an algorithm, which implements the method of τ-reduction for distinguishing among characters on κ^{−1}(c) when c < −14.

Algorithm τ-reduction(u) Input. A character u ∈ κ^{−1}(c). Output. A character in Ω_{0}^{M}, ℰ_{c}, or Σ_{0}^{c}, which is Γ-equivalent to u. u ← u_{0}; while |z̄| > 2 do if x(u) = 0 or y(u) = 0 then u ∈ Σ_{0}^{c} return u

if z(u) < −2 and z̄(u) < −2 then u ∈ Ω_{0}^{M} return u

if z(u) > 2 and z̄(u) > 2 then σ_{xz}(u) ∈ Ω_{0}^{M} return σ_{xz}(u)

if τ(Q_{x}(u)) < τ(u) then u ← Q_{x}(u)

elseif τ(Q_{y}(u)) < τ(u) then u ← Q_{y}(u)

else u ← Q_{z}(u)

end do

u ∈ ℰ_{c}; return u.

5. The Action of the Modular Group on Characters

In Chapter 3 we proved that the action of Γ on

Ω^{M} = ∐_{γ∈Γ} γΩ_{0}^{M}

is wandering. In this chapter we show that the action of Γ on the complement of Ω^{M} in 𝒳_{1,1} is ergodic in the following sense. Recall that the action of Γ induces a measurable equivalence relation “∼”, which is ergodic if and only if every function that is constant on equivalence classes is constant almost everywhere. In that context, if every point in a subspace X is Γ-equivalent to a point in a subspace Y, then ergodicity on X is equivalent to ergodicity on Y (regardless of whether Y is invariant or not; compare Goldman [8]). In the previous section we have proved that when c < 2 every point on (𝒳_{1,1} −Ω^{M} ∪ℰ_{c})∩κ^{−1}(c) is Γ-equivalent to a point in ℰ_{c}. Thus ergodicity on 𝒳_{1,1} reduces to ergodicity of “∼” on ℰ_{c}, or equivalently on its “Q_{z}-dual”

\overline{ℰ}_{c} = Q_{z}(ℰ_{c}) = \{(x, y, z) ∈ κ^{−1}(c) | −2 < z < 2\}

Let u = (x, y, z) ∈ \overline{ℰ}_{c}. Then e_{c}(z) = L_{z} ∩ κ^{−1}(c) is an ellipse upon which Q_{x}Q_{y} ∈ Λ_{x,y} acts by a rotation of angle

α = 2 cos^{−1} \frac{z}{2}.

(Compare Goldman [6].) For almost every z (namely when α/(2π) is irrational), this transformation generates an ergodic action on e_{c}(z). Thus a function

f : \overline{ℰ}_{c} ⟶ ℝ

that is Λ-invariant, would be constant almost everywhere on each L_{z} ∩ κ^{−1}(c) and would therefore depend almost everywhere on z alone. Thus f(x, y, z) = g(z) almost everywhere for some function

g : [−2, 2] ⟶ ℝ

It now suffices to eliminate the dependence of g on z. To this end, we parametrize e_{c}(z) as follows (see Lemma 4.5)

e_{c}(z) : \{ \begin{matrix}x & = \frac{\sqrt{2}}{2}(−A cos θ + B sin θ) \\ y & = \frac{\sqrt{2}}{2}( A cos θ + B sin θ) \end{matrix}

where

A = \sqrt{\frac{z^{2} − c − 2}{2 + z}}, B = \sqrt{\frac{z^{2} − c − 2}{2 − z}}

Consequently, for a fixed z ∈ (−2, 2) the restriction of z̄ to e_{c}(z) depends on θ alone

z̄_{c,z}(θ) = z̄|_{e_{c}(z)} = \frac{z^{2} − c − 2}{2 + z} cos^{2} θ − \frac{z^{2} − c − 2}{2 − z} sin^{2} θ − z

whose extrema are attained at θ = kπ/2, for k ∈ ℤ. The critical values of z̄_{c,z}(θ) are

(5.1)  \begin{aligned}z̄_{odd} & = −\frac{z^{2} − c − 2}{2 − z} − z & = \frac{−2z + c + 2}{2 − z} & = 2 + \frac{c − 2}{2 − z} \\ z̄_{even} & = \frac{z^{2} − c − 2}{2 + z} − z & = \frac{−2z − c − 2}{2 + z} & = −2 − \frac{c − 2}{2 + z} \end{aligned}

respectively, depending on the parity of k. We classify the extrema of z̄_{c,z} by analysing the second derivative

z̄′′_{c,z}(θ) = −\frac{8(z^{2} − c − 2)}{4 − z^{2}} cos 2θ

Clearly

z̄′′_{c,v} (\frac{kπ}{2} ) \{ \begin{matrix}> 0, & k odd \\ < 0, & k even \end{matrix}

provided that

(5.2)  z^{2} − c − 2 > 0

Observe that when c < −2, the latter is satisfied trivially for any z ∈ ℝ. On the other hand, when −2 < c < 2, inequality (5.2) is satisfied a fortiori for all points in

\overline{ℰ}_{c} = κ^{−1}(c) ∩ ℝ^{2} × (−2, 2)

since for any such point u = (x, y, z), the quadratic form

S_{z}(x, y) = −x^{2} − y^{2} + xyz

is negative definite and therefore the set

κ^{−1}(c) ∩ ℝ^{2} × (−\sqrt{c + 2}, \sqrt{c + 2})

is empty. Consequently, the critical value z̄_{odd} is a minimum, and the critical value z̄_{even} is a maximum of z̄_{c,z} for each z ∈ (−2, 2), such that L_{z} ∩ κ^{−1}(c) is non-empty. Clearly, these extrema are global. Thus for any z ∈ (−2, 2)

(5.3)  \begin{aligned}z̄_{min}(c, z) & = min_{θ∈ℝ} z̄_{c,z}(θ) & = 2 + \frac{c − 2}{2 − z} \\ z̄_{max}(c, z) & = max_{θ∈ℝ} z̄_{c,z}(θ) & = −2 − \frac{c − 2}{2 + z} \end{aligned}

When c < −14

(5.4)  (−2, 2) ⊊ (z̄_{min}(c, z), z̄_{max}(c, z))

for any z ∈ (−2, 2). The cyclic group generated by an irrational rotation of an ellipse acts ergodically and therefore, for almost z ∈ (−2, 2) and every u ∈ e_{c}(z), the set

{z̄(λu) | λ ∈ Λ_{x,y}}

must be dense in [z̄_{min}(c, z), z̄_{max}(c, z)]. Therefore the Q_{z}-invariance of f implies that f must be constant also with respect to z, for almost every z ∈ (−2, 2).

Notice that the set

Σ^{c} = Σ_{0}^{c} − (Σ_{0}^{c} ∩ ℰ_{c})

is invariant with respect to Q_{z}, σ_{xz}, σ_{yz} and transpositions of x and y. Clearly, Q_{x}Σ^{c} and Q_{y}Σ^{c} do not intersect ℰ_{c}. Therefore Γ · Σ^{c} ∩ ℰ_{c} = ∅. Moreover by definition |z(u)| > 2 for every u ∈ Σ^{c} and hence the set Λ_{x,y} lies on a hyperbola h_{c,z}. Recall that the action of Λ_{x,y} on h_{c,z} is wandering and therefore Λ_{x,y}u is nowhere dense in h_{c,z}. Consequently, the set

∐_{γ∈Γ} γΣ^{c}

has measure 0 in κ^{−1}(c). We now have a complete picture of the Γ-action on characters for the case when c < −14.

Theorem 5.1. Suppose c < −14. Then the action of Γ on the level sets κ^{−1}(c) is

(1) wandering on the set Ω^{M} of characters of discrete M-embeddings

(2) ergodic on the complement of Ω^{M} ∩ κ^{−1}(c)

κ^{−1}(c) ∩ (𝒳_{1,1} − Ω^{M})

Next, we extend this argument to the case when −14 < c < 2. In that case the inclusion (5.4) is no longer valid. However

z̄_{min}(c, z) < 2, z̄_{max}(c, z) > −2

for each z ∈ (−2, 2). Moreover, there exists a countable open cover of the interval (−2, 2) by intervals

I_{n} = (z̄_{min}(c, z_{n}), z̄_{max}(c, z_{n})) ∩ (−2, 2), n = 1, 2, …

(see Figure 14). In each I_{n} a Q_{z}-invariant function g(z) is constant almost everywhere

z̄_{max}(z, c)  z̄_{min}(z, c)
Figure 14: The functions z̄_{min}(z, c) and z̄_{max}(z, c)

by the argument above, and the values of g(z) agree by default on the overlaps I_{n} ∩ I_{n+1}. Therefore g(z) must be constant almost everywhere in (−2, 2).

Corollary 5.2. For any c < 2 the action of Γ on κ^{−1}(c) ∩ (𝒳_{1,1} − Ω^{M}) is ergodic.

We conclude with a remark about the special case c = −2. In that case the level set κ^{−1}(c) consists of characters u = (x, y, z) that satisfy the equation

−x^{2} − y^{2} + z^{2} + xyz = 0

The coordinates of such characters are closely related to Markoff triples, which play an important role in Number Theory. In a recent paper, Bowditch ([1]) studies the action of the modular group on complex Markoff triples and proves that the action has dense orbits in a neigborhood of the origin. Thus Bowditch’s result, restricted to real characters, is a special case of Corollary 5.2.

52 WILLIAM GOLDMAN AND GEORGE STANTCHEV

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Department of Mathematics, University of Maryland, College Park, MD 20742