THE TOPOLOGY OF THE RELATIVE CHARACTER

VARIETIES OF A QUADRUPLY-PUNCTURED SPHERE

ROBERT L. BENEDETTO AND WILLIAM M. GOLDMAN

Abstract. Let M be a quadruply-punctured sphere with boundary components A, B, C, D. The SL(2, ℂ)-character variety of M consists of equivalence classes of homomorphisms ρ of π_{1}(M) ⟶ SL(2, ℂ) and can be identified with a quartic hypersurface in ℂ^{7}. For fixed a, b, c, d ∈ ℂ, the subset V_{a,b,c,d} corresponding to representations ρ with tr(ρ(A)) = a, tr(ρ(B)) = b, tr(ρ(C)) = c, tr(ρ(D)) = d is a cubic surface in ℂ^{3}. We determine the singular points of V_{a,b,c,d}, classify its set V_{a,b,c,d}(ℝ) of ℝ-points into six topological types, at least when V_{a,b,c,d}(ℝ) is nonsingular. V_{a,b,c,d}(ℝ) contains a compact component if and only if −2 < a, b, c, d < 2. For certain (a, b, c, d), this component corresponds to representations in SL(2, ℝ).

Contents

1. Introduction 1 2. Character varieties for surfaces 4 2.1. Cyclic groups 4 2.2. The triply-punctured sphere 7 2.3. Properties of κ(x, y, z) 8 2.4. The quadruply-punctured sphere 9 2.5. Amalgamating representations 9 2.6. Defining equations 10 2.7. Symmetries 11 3. Singular points and reducible characters 13 3.1. A criterion for singularity 13

Date: October 26, 1997. 1991 Mathematics Subject Classification. 57M05 (Low-dimensional topology). Key words and phrases. Surface, Fundamental group, Character variety, Repre-

sentation variety.

Goldman gratefully acknowledges partial support from National Science Foundation grants DMS-8613576, DMS-8902619, DMS-9205139, DMS-9504764, the Alfred P. Sloan Foundation, and the University of Maryland Institute for Advanced Computer Studies.

3.2. Solutions of quadratic equations 13 3.3. Singularities of V_{a,b,c,d} 15 3.4. A discriminant polynomial 17 4. Topology of the set of ℝ-points 18 4.1. Singular points 18 4.2. Topological classification of the smooth real varieties 20 4.3. Reduction to a plane quartic curve 20 4.4. The six cases 26 5. A compact component of SL(2, ℝ)-representations 32 References 36 List of Figures 36

1. Introduction

Moduli spaces of representations of fundamental groups of surfaces arise in many natural algebraic, geometric, and analytic problems. In particular, the classification of geometric structures on manifolds and solutions of the Yang-Mills equations from gauge theory lead to moduli spaces closely related to representations of surface groups. Furthermore these moduli spaces enjoy a rich symmetry due to the large topological symmetry groups of surfaces. The present paper describes in detail the topology of some of these moduli spaces, in the case of surfaces homeomorphic to a quadruply-punctured sphere.

Decompositions of surfaces into subsurfaces provide a technique for analyzing the moduli spaces in terms of moduli spaces for simpler surfaces. For example, every compact connected surface M of genus g with b boundary components decomposes along 3g +b−3 simple closed curves into 2g + b − 2 “pants”, provided that

0 > χ(M) = 2 − 2g − b.

A “pants” is a triply-punctured sphere, that is, a surface of genus 0 with three boundary components. In general, we shall refer to a connected surface of genus g with b boundary components as a b-punctured surface of genus g, although, strictly speaking the ends of a “punctured surface” are noncompact and do not have boundary. Similarly, a bpunctured torus is a surface of genus 1 with b boundary components. Surfaces of small negative Euler characteristic are the building blocks for all surfaces of negative Euler characteristic and their moduli spaces are the building blocks for moduli spaces of more complicated surfaces.

This decomposition technique has been used in [7] to study the automorphism group of surfaces and in [5] to analyze moduli spaces of geometric structures on surfaces.

This paper concerns the moduli space of representations of a quadruply-punctured sphere into SL(2, ℂ). This moduli space is naturally parametrized by the moduli of the images of the boundary components, which are described by four scalar parameters (a, b, c, d). For fixed values of (a, b, c, d), the moduli space is a surface V_{a,b,c,d}. We determine for which (a, b, c, d) is V_{a,b,c,d} nonsingular.

Theorem. For fixed a, b, c, d ∈ ℂ, the surface V_{a,b,c,d} singular if and only if either at least one of a, b, c, d equals ±2 or there is a reducible representation ρ with boundary traces a, b, c, d. The latter case occurs when Δ(a, b, c, d) = 0 where Δ(a, b, c, d) is the polynomial

Δ(a, b, c, d) = (2(a^{2} + b^{2} + c^{2} + d^{2}) − abcd − 16)^{2}
− (4 − a^{2})(4 − b^{2})(4 − c^{2})(4 − d^{2})

In this case V_{a,b,c,d}(ℂ) is a cubic surface in ℂ^{3} whose projective completion V̄_{a,b,c,d}(ℂ) is smooth. Furthermore V̄_{a,b,c,d}(ℂ) \ V_{a,b,c,d} consists of three lines in general position, and using the classification of cubic surfaces, one proves that the set of ℂ-points of V_{a,b,c,d} deformation retracts to a bouquet of five 2-spheres.

However, the topologies of the real varieties are much more varied. For (a, b, c, d) ∈ ℝ, the set V_{a,b,c,d}(ℝ) of ℝ-points of V_{a,b,c,d}, at least when V_{a,b,c,d}(ℝ) is nonsingular, one of six topological types, depending on the boundary parameters a, b, c, d. These six topological types are depicted in Figures 11, 12, 13, 14, 15, and 16. Let n denote the number of boundary traces a, b, c, d which lie in the interval (−2, 2):

Theorem. Let a, b, c, d ∈ ℝ be such that V_{a,b,c,d} is nonsingular. Then the set V_{a,b,c,d}(ℝ) of ℝ-points is homeomorphic to one of the following:

1. A quadruply-punctured sphere if n = 0 and abcd < 0;

2. A disjoint union of a triply-punctured torus and a disc if n = 0 and abcd > 0;

3. A disjoint union of a triply-punctured sphere and a disc if n = 1;

4. A disjoint union of an annulus and two discs if n = 2;

5. A disjoint union of four discs if n = 3;

6. A disjoint union of four discs and a sphere if n = 4.

Corollary. For a, b, c, d and n as above, the Euler characteristic of V_{a,b,c,d}(ℝ) is given by the following formula:

χ(V_{a,b,c,d}(ℝ)) = 2n − 2

The points of V_{a,b,c,d}(ℝ) correspond to representations into either SU(2) or SL(2, ℝ). Components of spaces of representations into SU(2) are always compact, and in the analogous case where M is a oncepunctured torus, the only compact components of relative real character varieties of M correspond to SU(2)-representations. When −2 < a, b, c, d < 2, the variety V_{a,b,c,d}(ℝ) contains a compact component, and we determine conditions when this compact component corresponds to representations in SU(2) or SL(2, ℝ).

Proposition. Suppose a, b, c, d ∈ (−2, 2) and V_{a,b,c,d} is smooth. Then V_{a,b,c,d}(ℝ) contains a compact component, which consists of characters of SL(2, ℝ)-representations if and only if

Δ(a, b, c, d) > 0
16 − abcd − 2(a^{2} + b^{2} + c^{2} + d^{2}) > 0.

Otherwise, each element in the component is the character of an SU(2)- representation.

The component of V_{a,b,c,d}(ℝ) homeomorphic to a disc corresponds to the “Teichmüller space” of M and consists of characters of holonomy representations of hyperbolic structures on M with geodesic boundary. In all of the last five cases, the component which is homeomorphic to a disc admits a similar interpretation in terms of hyperbolic structures with conical singularities; see [5].

This investigation began as “Research Experiences for Undergraduates” project at the University of Maryland in the summers of 1992– 1993, under Goldman’s direction. During this project, Benedetto used Geomview [4] and Mathematica [10] software on Silicon Graphics workstations to visualize the components of the real moduli spaces. Many of the results in this paper were suggested by computer experiments before they were rigorously proved. (Compare Benedetto’s senior thesis [1]). The illustrations in this paper were drawn using Geomview and Mathematica software on Silicon Graphics and NeXT workstations.

2. Character varieties for surfaces

Let k denote a field (usually either ℝ or ℂ) and let G = SL(2, k). For the moment let π denote an arbitrary finitely generated group. We denote the set of homomorphisms π ⟶ G by Hom(π, G); since G is an affine algebraic group defined over k, Hom(π, G) naturally enjoys the structure of the set of k-points of an affine algebraic set defined over ℤ. (Compare Lubotzky-Magid [8],§1.) A central problem is to determine the global topology and geometric structure on the space Hom(π, G).

We would like to consider the space of equivalence classes of such representations under the natural action of SL(2, ℂ) by conjugation, but the quotient space is generally not well-behaved. By enlarging the equivalence relation slightly, one can define a categorical quotient Hom(π, G)//G (in the sense of algebraic geometry) which does have the structure of an affine algebraic set. For brevity denote Hom(π, G) by X. The guiding principle is that the regular functions on the quotient X//G should be the G-invariant regular functions on X, and thus the quotient X//G is defined as Spec(k[X]). The points of X//G correspond to equivalence classes, where two orbits are considered equivalent if they cannot by distinguished by G-invariant regular functions on X. In general there is a G-invariant Zariski-dense subset X^{−} such that the restriction of the quotient map X ⟶ X//G to X^{−} defines a bijection of the orbit space X^{−}/G to X//G. In practice, we can take (assuming k is algebraically closed) X^{−} to be the set of completely reducible (or semisimple) representations π ⟶ G. For a careful treatment of this material, we refer the reader to Lubotzky-Magid [8],§1.

2.1. Cyclic groups. When π is a cyclic group, then a homomorphism π ⟶ G is completely determined by the image of a generator of π, and every regular function on G which is invariant under conjugation is a polynomial in the trace function

G ⟶ k
A ⟼ tr(A)

Thus X ≅ k under the trace mapping. Note that the trace function has critical values ±2, with critical points at the center ±I.

More generally, suppose that π is a free group freely generated by x_{1}, …, x_{n}. Then Hom(π, G) identifies with the Cartesian product G^{n} and Procesi [12] proved that every G-invariant function on G^{n} is a polynomial in finitely many functions of the form

G^{n} ⟶ k
(A_{1}, …, A_{n}) ⟼ tr w(A_{1}, …, A_{n})

where w(x_{1}, …, x_{n}) is a word in x_{1}, …, x_{n}, that is, an element of π.

For example, for n = 2 and k = ℂ, classical work (dating at least as far back as Fricke and Klein [3]) implies the following fundamental result:

Theorem 2.1.1. The invariant function

χ : Hom(π, G) ⟶ ℂ^{3}
ρ ⟼ [\begin{matrix}tr(ρ(x_{1})) \\ tr(ρ(x_{2})) \\ tr(ρ(x_{1}x_{2})) \end{matrix}]

defines an isomorphism X ⟶ ℂ^{3}. That is, every function of pairs (A_{1}, A_{2}) ∈ SL(2, ℂ) × SL(2, ℂ) invariant under simultaneous conjugation

(A_{1}, A_{2}) ↦ (T A_{1}T^{−1}, T A_{2}T^{−1})

can be expressed is a function of tr(A_{1}), tr(A_{2}), and tr(A_{1}A_{2}). Furthermore, if A_{1}, A_{2} generate an irreducible representation on ℂ^{2}, then for any other pair A′_{1}, A′_{2} with

tr(A_{1}) = tr(A′_{1})
tr(A_{2}) = tr(A′_{2})
tr(A_{1}A_{2}) = tr(A′_{1}A′_{2})

there exists T ∈ SL(2, ℂ) with A′_{1} = T A_{1}T^{−1}, A′_{2} = T A_{2}T^{−1}.

We call the triple χ(ρ) the character of ρ, since from it one can deduce the trace tr(ρ(A)) for any A ∈ π. (Compare Magnus [9].)

The criterion for reducible representations is slightly more complicated. If ρ is reducible, then it is conjugate to a representation in upper-triangular matrices

ρ(γ) = [\begin{matrix}a(γ) & b(γ) \\ 0 & a(γ)^{−1} \end{matrix}]

It and its semisimplification

ρ^{s}(γ) = [\begin{matrix}a(γ) & 0 \\ 0 & a(γ)^{−1} \end{matrix}]

have the same character, but in general will not be conjugate. Indeed, ρ^{s} lies in the closure of the orbit of ρ. Two reducible representations have the same character if and only if their semisimplifications are conjugate, or equivalently, if their orbit closures are equal.

For example, the G-invariant function on G × G defined by

f : (A_{1}, A_{2}) ↦ tr(A_{1}A_{2}A_{1}^{−1}A_{2}^{−1})

is given by the expression

f(A_{1}, A_{2}) = κ(tr(A_{1}), tr(A_{2}), tr(A_{1}A_{2}))

where κ ∈ ℂ[x, y, z] is the cubic polynomial defined by:

κ(x, y, z) = x^{2} + y^{2} + z^{2} − xyz − 2 (2.1.1)

A slight variant of this polynomial will be useful later on. Let a, b ∈ k. Then κ_{a,b}(x) ∈ k[x] is the quadratic polynomial defined by:

κ_{a,b}(x) = κ(a, b, x) − 2 = x^{2} − abx + (a^{2} + b^{2} − 4) (2.1.2)

The reducible representations occur for characters (x, y, z) with

κ(x, y, z) = 2

where κ : ℂ^{3} ⟶ ℂ is defined by (2.1.1). We refer to such a character as a reducible character.

The set of ℝ-points in 𝔛(M) equals 𝔛(M)(ℝ) = ℝ^{3}. By Prop. III.1.1 of Morgan-Shalen [11] (p.458), real characters correspond to representations taking values in the real forms SU(2) and SL(2, ℝ) of SL(2, ℂ). A point (x, y, z) ∈ ℝ^{3} corresponds to an SU(2)-representation if and only if

−2 ≤ x, y, z ≤ 2 and κ(x, y, z) ≤ 2.

Otherwise (x, y, z) ∈ ℝ^{3} corresponds to an SL(2, ℝ)-representation. A character (x, y, z) ∈ ℝ^{3} corresponds to both an SU(2)-representation and an SL(2, ℝ)-representation if and only if it corresponds to a representation in

SO(2) = SU(2) ∩ SL(2, ℝ);

such a representation is necessarily reducible. See Goldman [6] for details.

2.2. The triply-punctured sphere. We interpret Theorem 2.1.1 in terms of the the triply-punctured sphere. For any compact manifold, denote

𝔛(M) = ∏_{i} Hom(π_{1}(M_{i}), G)//G

where

M = ∐_{i} M_{i}

is the decomposition of M into connected components. When M is a compact surface with boundary

∂M = ∐_{i=1}^{m} ∂_{i}M

each component ∂_{i}M has fundamental group π_{1}(∂_{i}M) ≅ ℤ. Thus each 𝔛(∂_{i}M) ≅ ℂ and 𝔛(∂M) ≅ ℂ^{m}. The homomorphism of fundamental groups

π_{1}(∂_{i}M) ⟶ π_{1}(M)

induces a restriction map

𝔛(M) ⟶ 𝔛(∂_{i}M) ≅ ℂ.

Taking the product, we obtain

∂_{∗} : 𝔛(M) ⟶ ℂ^{m}

Let M denote a triply-punctured sphere. Its fundamental group admits the redundant geometric presentation

π = ⟨X, Y, Z | XY Z = I⟩

where X, Y, Z are simple loops correponding to the three boundary components. In this case the boundary map ∂_{∗} defines an isomorphism 𝔛(M) ≈ ℂ^{3}, where

∂_{∗}([ρ]) = (tr(ρ(X)), tr(ρ(Y)), tr(ρ(Z))).

2.3. Properties of κ(x, y, z). The polynomial κ and its variant κ_{a,b} satisfy many remarkable properties related to their natural occurence as invariants of elements of SL(2, ℂ).

We use the following lemma concerning κ_{a,b} and ℝ-points extensively in later sections. Its proof is straightforward.

Figure 1. A triply-punctured sphere and its three boundary components Lemma 2.3.1. Fix a, b ∈ ℝ. Then κ_{a,b} has:

• One real double root if either of a or b is ±2;

• No real roots if |a| > 2 and |b| < 2 or vice versa;

• Two real roots, both greater than or equal to 2, if |a|, |b| > 2 and ab > 0;

• Two real roots, both less than or equal to −2, if |a|, |b| > 2 and ab < 0;

• Two real roots, both in [−2, 2], if |a|, |b| < 2.

This lemma illustrates the properties of reducible characters, since x is a root of κ_{a,b} if and only if (a, b, x) is a reducible character. Indeed, we may assume that (a, b, x) is the character of a representation with abelian image. For example, the second case occurs when a is the trace of a hyperbolic element and b is the trace of an elliptic element, and in SL(2, ℝ) hyperbolic elements and elliptic elements cannot commute. The other cases can be discussed similarly.

An important property of κ is that its restriction to every coordinate plane and coordinate line is quadratic. In particular its values at ±2 are very special.

Lemma 2.3.2. κ_{a,b}(±2) = (a ± b)^{2}.

The discriminant of the quadratic polynomial κ_{a,b}(x) equals

(a^{2} − 4)(b^{2} − 4).

Furthermore the critical points of κ : ℂ^{3} ⟶ ℂ are the four points

(2, 2, 2),  (2, −2, −2),  (−2, 2, −2),  (−2, −2, 2)

with critical value +2 and the point (0, 0, 0) with critical value −2.

2.4. The quadruply-punctured sphere. Let M be a quadruplypunctured sphere with boundary components A, B, C, D. Denoting the corresponding elements of π = π_{1}(M) by the same name, the fundamental group π = π_{1}(M) has the redundant geometric presentation

π = ⟨A, B, C, D | ABCD = I⟩

(Compare Figure 2.) However, in the present context there is an even more concrete description. A representation ρ : π ⟶ SL(2, ℂ) corresponds to a quadruple

α = ρ(A), β = ρ(B),
γ = ρ(C), δ = ρ(D)

in SL(2, ℂ)^{4} where

αβγδ = I,

Figure 2. A quadruply-punctured sphere

Using this equation to eliminate δ, such a quadruple merely refers to an arbitrary triple (α, β, γ) ∈ SL(2, ℂ)^{3}.

2.5. Amalgamating representations. Let ξ denote the simple loop on M which separates boundary components A, B from C, D. The corresponding element of the fundamental group satisfies:

X = AB = (CD)^{−1}.

Denote by M|ξ the compact surface obtained by splitting M along ξ: each of the two components M_{1} and M_{2} of M|ξ are homeomorphic to a triply-punctured sphere. We express the character variety 𝔛(M) in terms of the character variety 𝔛(M|ξ).

The fundamental group of M_{1} has presentation

π_{1}(M_{1}) = ⟨A, B, X̄ | ABX̄ = I⟩

and the fundamental group of M_{2} has presentation

π_{1}(M_{2}) = ⟨X, C, D | XCD = I⟩.

The fundamental group of M is the amalgamated free product

π_{1}(M) = π_{1}(M_{1}) ∐_{ℤ}π_{1}(M_{2})

amalgamated by the monomorphisms taking a generator of ℤ to X̄^{−1} ∈ π_{1}(M_{1}) and X ∈ π_{1}(M_{2}) respectively. A representation

ρ : π_{1}(M) ⟶ SL(2, ℂ)

restricts to two representations

ρ_{i} : π_{1}(M_{i}) ⟶ SL(2, ℂ)

satisfying

ρ_{1}(X̄)ρ_{2}(X) = I. (2.5.1)

Conversely two representations ρ_{1}, ρ_{2} satisfying (2.5.1) define a unique representation ρ. (Compare Goldman [5].)

2.6. Defining equations. The boundary map ∂_{∗} : 𝔛(M) ⟶ ℂ^{4} is defined by four functions

a([ρ]) = tr(ρ(A)), b([ρ]) = tr(ρ(B)),
c([ρ]) = tr(ρ(C)), d([ρ]) = tr(ρ(D))

The elements X = AB, Y = BC and Z = CA of π are represented by simple loops on M. The corresponding traces x, y, z, together with the boundary traces a, b, c, d, define coordinates on 𝔛(M) which are subject to the relation

x^{2} + y^{2} + z^{2} + xyz = (ab + cd)x
+ (ad + bc)y + (ac + bd)z

+ (4 − a^{2} − b^{2} − c^{2} − d^{2} − abcd) (2.6.1)

(Compare Magnus [9].) Unlike the case in which π is free of rank two, 𝔛(M) is not an affine space, but rather the hypersurface V ⊂ ℂ^{7} consisting of points whose coordinates (a, b, c, d, x, y, z) satisfy (2.6.1). For each (a, b, c, d) ∈ ℂ^{4}, the set

∂_{∗}^{−1}(a, b, c, d) = V_{a,b,c,d}

consists of all (x, y, z) ∈ ℂ^{3} satisfying (2.6.1). Rewrite the basic defining equation (2.6.1) in terms of x as follows:

\frac{2 + x}{4}((y + z) − \frac{(a + b)(d + c)}{2 + x})^{2}

+ \frac{2 − x}{4}((y − z) − \frac{(a − b)(d − c)}{2 − x})^{2}

= \frac{(x^{2} − abx + a^{2} + b^{2} − 4)(x^{2} − cdx + c^{2} + d^{2} − 4)}{4 − x^{2}} (2.6.2)

We may write (2.6.2) as follows. For fixed a, b, c, d, x,

Q_{x}^{(a,b,c,d)}(y, z) =

\frac{2 + x}{4}((y + z) − \frac{(a + b)(d + c)}{2 + x})^{2}

+ \frac{2 − x}{4}((y − z) − \frac{(a − b)(d − c)}{2 − x})^{2}

is an inhomogeneous quadratic function of (y, z). Equation (2.6.2) is equivalent to:

Q_{x}^{(a,b,c,d)}(y, z) = \frac{κ_{a,b}(x)κ_{c,d}(x)}{4 − x^{2}}

which describes a conic V_{a,b,c,d}(x) centered at (y_{0}, z_{0}) where

y_{0} + z_{0} = \frac{(a + b)(d + c)}{2 + x}
y_{0} − z_{0} = \frac{(a − b)(d − c)}{2 − x}

2.7. Symmetries. A group of 192 symmetries of the family 𝔛(M) acts by simple linear transformations on the parameters (a, b, c, d). This action is generated by an action of the central characters, which multiply an even number of the generators by −I, and the permutations of the four generators A, B, C, D.

The symmetric group S_{4} consisting of permutations of {A, B, C, D} acts on the family 𝔛(M) of relative character varieties V_{a,b,c,d}. The symmetries are apparent from the defining equation (2.6.1). For example the three products of disjoint transpositions act trivially on the (x, y, z) coordinates, while nontrivially permuting (a, b, c, d). However, the transpositions A ↔ B, B ↔ C, C ↔ D, act as follows:

[\begin{matrix}a \\ b \\ c \\ d \\ x \\ y \\ z \end{matrix}] ↦ [\begin{matrix}b \\ a \\ c \\ d \\ x \\ z \\ y \end{matrix}] ,  [\begin{matrix}a \\ b \\ c \\ d \\ x \\ y \\ z \end{matrix}] ↦ [\begin{matrix}a \\ c \\ b \\ d \\ z \\ y \\ x \end{matrix}] ,  [\begin{matrix}a \\ b \\ c \\ d \\ x \\ y \\ z \end{matrix}] ↦ [\begin{matrix}a \\ b \\ d \\ c \\ x \\ z \\ y \end{matrix}]

Furthermore every character variety admits an action by central characters, as described in Lubotzky-Magid [8], §5 (where this operation is called twisting by a character.) If Z ⊂ G is central, then Hom(π, Z) is a group, which acts on Hom(π, G) by pointwise multiplication. Since the center of G = SL(2, ℂ) equals {±I} ≅ ℤ/2 and the fundamental group π of a quadruply-punctured sphere is free of rank 3, then

Hom(π, {±I}) ≅ ℤ/2 ⊕ ℤ/2 ⊕ ℤ/2

acts on V_{a,b,c,d} as follows. Such a central character maps the images of an even number of the generators A, B, C, D to their negatives: for example,

[\begin{matrix}α \\ β \\ γ \\ δ \end{matrix}] ↦ [\begin{matrix}−α \\ −β \\ γ \\ δ \end{matrix}]

is the character corresponding to

(1, 1, 0) ∈ ℤ/2 ⊕ ℤ/2 ⊕ ℤ/2

and whose action on the trace coordinates is given by:

[\begin{matrix}a \\ b \\ c \\ d \\ x \\ y \\ z \end{matrix}] ↦ [\begin{matrix}−a \\ −b \\ c \\ d \\ x \\ −y \\ −z \end{matrix}]

S_{4} normalizes this Hom(π, {±I}) - action. Since the induced S_{4}-action on Hom(π, {±I}) is transitive, the action of every central character is obtained by conjugating the above example by a permutation. These considerations of symmetry immediately imply the corresponding relative character varieties are isomorphic:

Lemma 2.7.1. Let (a, b, c, d) ∈ ℝ^{4}. Let σ : ℝ^{4} → ℝ^{4} be any permutation of coordinates, and let τ : ℝ^{4} → ℝ^{4} be any map which acts by reversing signs of two of the coordinates and leaving the other two coordinates unchanged. Then V_{σ(a,b,c,d)} and V_{τ(a,b,c,d)} are both diffeomorphic to V_{a,b,c,d}.

3. Singular points and reducible characters

3.1. A criterion for singularity.

Theorem 3.1.1. V_{a,b,c,d} is singular if and only if at least one of the following holds:

• One of a, b, c, d equals ±2;

• Some reducible representation has boundary traces (a, b, c, d). The latter case occurs if and only if there exists x ∈ ℂ such that κ_{a,b}(x) = κ_{c,d}(x) = 0.

3.2. Solutions of quadratic equations. We will need the following technical lemmas on discriminant loci for the proof of Theorem 3.1.1 and for later results.

Lemma 3.2.1. Let n ≥ 1 be an integer, and let k be a field of characteristic not equal to 2. Pick B(x), C(x) ∈ k[x_{1}, …, x_{n}]. Let

Y = {(x_{1}, …, x_{n}, z) ∈ k^{n+1} | z^{2} + B(x)z + C(x) = 0}

and

Z = {(x_{1}, …, x_{n}) ∈ k^{n} | B(x)^{2} − 4C(x) = 0}.

If Y is singular at (x_{1}, …, x_{n}, z), then Z is singular at (x_{1}, …, x_{n}). If Z is singular at (x_{1}, …, x_{n}), then there exists z ∈ k such that Y is singular at (x_{1}, …, x_{n}, z).

(If n = 1, then Z is singular at x if x is a multiple root of B(x)^{2}−4C(x).)

Proof. Denote by dB(x) the differential of B(x) with respect to the x_{i}-variables only, and similarly for dC(x). Let (x_{1}, …, x_{n}) ∈ k^{n}. Now Z is singular at (x_{1}, …, x_{n}) if and only if

\begin{cases}B(x)^{2} − 4C(x) & = 0 \\ 2B(x) dB(x) − 4 dC(x) & = 0 \end{cases} (3.2.1)

and Y is singular at (x_{1}, …, x_{n}, z) if and only if

\begin{cases}z^{2} + B(x)z + C(x) & = 0 \\ 2z + B(x) & = 0 \\ z dB(x) + dC(x) & = 0. \end{cases} (3.2.2)

We must show that (3.2.1) is equivalent to the existence of z ∈ k satisfying (3.2.2). Note that

B(x)^{2} − 4C(x) = 0

if and only if

z^{2} + B(x)z + C(x)

has a double root z ∈ k; necessarily this root must be

z = −\frac{1}{2}B(x).

Substituting z = −\frac{1}{2}B(x) into the final equation of (3.2.2) and multiplying by −4 yields the second equation of (3.2.1).

Lemma 3.2.2. Let k be an algebraically closed field of characteristic not equal to 2. Suppose A_{1}(x), B_{1}(x), C_{1}(x) ∈ k[x] such that A_{1}(x) has no multiple roots. Let

Y = {(x, y) ∈ k^{2} | A_{1}(x)y^{2} + B_{1}(x)y + C_{1}(x) = 0}

and

Z = {x ∈ k | B_{1}(x)^{2} − 4A_{1}(x)C_{1}(x) = 0}.

If Y is singular at (x, y), then Z is singular at x. If Z is singular at x, then there exists y ∈ k such that Y is singular at (x, y).

Proof. The proof follows that of Lemma 3.2.1, modified in the case that A_{1}(x) = 0. Suppose that A_{1}(x) = 0. Then Z is singular at x if and only if

\begin{cases}B_{1}(x)^{2} & = 0 \\ 2B_{1}(x)B′_{1}(x) − 4C_{1}(x)A′_{1}(x) & = 0 \end{cases} (3.2.3)

and Y is singular at (x, y) if and only if there exists y ∈ k such that

\begin{cases}yB_{1}(x) + C_{1}(x) & = 0 \\ B_{1}(x) & = 0 \\ y^{2}A′_{1}(x) + yB′_{1}(x) + C′_{1}(x) & = 0 \end{cases} (3.2.4)

Clearly (3.2.3) is equivalent to B_{1}(x) = C_{1}(x) = 0, which (3.2.4) implies. Since A′_{1}(x) ≠ 0 (by hypothesis), and k is algebraically closed, some z ∈ k satisfies the last equation in (3.2.4), concluding the proof of Lemma 3.2.2.

Lemma 3.2.3. Let k = ℝ. Suppose A_{1}(x), B_{1}(x), C_{1}(x) ∈ ℝ[x] such that A_{1}(x) has no multiple roots. Let

Y = {(x, y) ∈ ℝ^{2} | A_{1}(x)y^{2} + B_{1}(x)y + C_{1}(x) = 0}
Z = {x ∈ ℝ | B_{1}(x)^{2} − 4A_{1}(x)C_{1}(x) = 0}.

and let

δ(x) = B′_{1}(x)^{2} − 4A′_{1}(x)C′_{1}(x). (3.2.5)

If Y is singular at (x, y), then Z is singular at x. Suppose for some x ∈ ℝ that

A_{1}(x) = B_{1}(x) = C_{1}(x) = 0

and

δ(x) ≥ 0

If Z is singular at x, then there exists y ∈ ℝ such that Y is singular at (x, y).

Proof. The only point at which the proof of Lemma 3.2.2 fails for k = ℝ is the final step, where the last equation in (3.2.4) may not have a solution y ∈ ℝ. Since the discriminant of this quadratic equation equals δ(x), our assumption δ(x) ≥ 0 guarantees the existence of a solution y ∈ ℝ.

3.3. Singularities of V_{a,b,c,d}.

Proof of Theorem 3.1.1. Fix (a, b, c, d) ∈ ℂ^{4}, and rewrite (2.6.1) as:

z^{2} + B(x, y)z + C(x, y) = 0

where

B(x, y) = xy − (ac + bd)
(3.3.1)  C(x, y) = x^{2} + y^{2} − (ab + cd)x − (ad + bc)y
− (4 − a^{2} − b^{2} − c^{2} − d^{2} − abcd).

By Lemma 3.2.1, V_{a,b,c,d} is singular if and only if the curve C in the xy-plane defined by the discriminant

f(x, y) = B(x, y)^{2} − 4C(x, y)
= x^{2}y^{2} − 4x^{2} − 4y^{2}

− 2(ac + bd)xy + 4(ab + cd)x + 4(ad + bc)y (3.3.2)

+ a^{2}c^{2} + b^{2}d^{2} − 2abcd
− 4a^{2} − 4b^{2} − 4c^{2} − 4d^{2} + 16 = 0

is singular. Now write f(x, y) as a quadratic function in y:

f(x, y) = A_{1}(x)y^{2} + B_{1}(x)y + C_{1}(x) (3.3.3)

where

A_{1}(x) = (x^{2} − 4)
B_{1}(x) = 4(ad + bc) − 2(ac + bd)x
(3.3.4)  C_{1}(x) = −4x^{2} + 4(ab + cd)x +
a^{2}c^{2} + b^{2}d^{2} − 2abcd
− 4a^{2} − 4b^{2} − 4c^{2} − 4d^{2} + 16 = 0.

Since A_{1}(x) = (x − 2)(x + 2) has simple zeroes, Lemma 3.2.2 implies that V_{a,b,c,d} is singular if and only if the discriminant g(x) of f(x, y) as a quadratic function of y vanishes. This discriminant is easily computed to be:

g(x) = B_{1}(x)^{2} − 4A_{1}(x)C_{1}(x)

= 16(x^{2} − abx + a^{2} + b^{2} − 4)(x^{2} − cdx + c^{2} + d^{2} − 4) (3.3.5)

= 16(κ_{a,b}(x))(κ_{c,d}(x)).

Thus V_{a,b,c,d} is singular if and only if there exists x ∈ k satisfying one of the following three conditions:

1. x is a double zero of κ_{a,b}; 2. x is a double zero of κ_{c,d};

3. κ_{a,b}(x) = κ_{c,d}(x) = 0. Now the first case occurs if and only if the discriminant of κ_{a,b} is zero, that is, if and only if

(a^{2} − 4)(b^{2} − 4) = 0,

that is, if and only if a = ±2 or b = ±2. Similarly, the second case occurs if and only if c = ±2 or d = ±2.

Consider the remaining case

κ_{a,b}(x) = κ_{c,d}(x) = 0 (3.3.6)

We must find a reducible representation having traces (a, b, c, d; x).

Decompose M along ξ as above into two triply-punctured spheres M_{1} and M_{2}. (Compare §2.5.) Then there exist representations

ρ_{i} : π_{1}(M_{i}) ⟶ SL(2, ℂ)

having characters (a, b, x) and (x, c, d) respectively. By (3.3.6), both ρ_{1} and ρ_{2} are reducible.

We distinguish two cases, depending on whether x = ±2 or not. If x = ±2, then choose representations ρ_{1}, ρ_{2} so that ρ_{1}(X̄) ≠ ±I and ρ_{2}(X) ≠ ±I. Necessarily ρ_{1}(X̄) and ρ_{2}(X) have unique fixed points p_{1} and p_{2}, respectively, in ℂℙ^{1}. There exists g ∈ G such that

g(p_{1}) = p_{2}, gρ_{1}(X̄^{−1})g^{−1} = ρ_{2}(X). (3.3.7)

Define ρ by:

ρ(A) = ρ_{1}(A)  ρ(B) = ρ_{1}(B)
(3.3.8)  ρ(C) = gρ_{2}(C)g^{−1}  ρ(D) = gρ_{1}(D)g^{−1}

Since the image of ρ fixes p_{1}, it is reducible and its character lies in V_{a,b,c,d}. (Alternatively, one can choose

ρ(X̄) = ±I

and ρ(C), ρ(D) to be diagonal matrices.)

Now suppose x ≠ ±2. Then ρ_{1}(X̄^{−1}) and ρ_{2}(X) are elements of G having trace x and are thus conjugate. Indeed, they are diagonalizable with with distinct eigenvalues λ, λ^{−1} where x = λ + λ^{−1}. Let p_{1} ∈ ℂℙ^{1} be the fixed point of ρ_{1}(X̄^{−1}) corresponding to the λ-eigenspace of ρ_{1}(X̄^{−1}) and p_{2} ∈ ℂℙ^{1} be the fixed point of ρ_{2}(X^{−1})) corresponding to the λ-eigenspace of ρ_{2}(X^{−1})). Then as above there exists g satisfying (3.3.7). The representation ρ defined by (3.3.8) fixes p_{1} and its character lies in V_{a,b,c,d}.

3.4. A discriminant polynomial. Given (a, b, c, d) ∈ ℂ^{4}, the existence of an x ∈ ℂ satisfying condition (3.3.6) is equivalent to the vanishing of the resultant Δ(a, b, c, d) of the polynomials κ_{a,b} and κ_{c,d}, which (remarkably) is a symmetric polynomial in a, b, c, d. This resultant is the sextic polynomial

Δ(a, b, c, d) = |\begin{matrix}1 & −ab & a^{2} + b^{2} − 4 & 0 \\ 0 & 1 & −ab & a^{2} + b^{2} − 4 \\ 1 & −cd & c^{2} + d^{2} − 4 & 0 \\ 0 & 1 & −cd & c^{2} + d^{2} − 4 \end{matrix}|

= a^{2}b^{2}c^{2} + a^{2}b^{2}d^{2} + a^{2}c^{2}d^{2} + b^{2}c^{2}d^{2}
− a^{3}bcd − ab^{3}cd − abc^{3}d − abcd^{3}

+ a^{4} + b^{4} + c^{4} + d^{4} + 8abcd (3.4.1)

− 2(a^{2}b^{2} + a^{2}c^{2} + a^{2}d^{2} + b^{2}c^{2} + b^{2}d^{2} + c^{2}d^{2})
= (2(a^{2} + b^{2} + c^{2} + d^{2}) − abcd − 16)^{2}
− (4 − a^{2})(4 − b^{2})(4 − c^{2})(4 − d^{2}).

Note that the vanishing of Δ(a, b, c, d) implies only that at least one character in V_{a,b,c,d} is reducible. In general, a character

(a, b, c, d, x, y, z) ∈ V_{a,b,c,d}

is reducible if and only if

κ_{a,b}(x) = κ_{c,d}(x) =
κ_{a,d}(y) = κ_{b,c}(y) =
κ_{a,c}(z) = κ_{b,d}(z) = 0

4. Topology of the set of ℝ-points

4.1. Singular points. For (a, b, c, d) ∈ ℝ^{4}, the variety V_{a,b,c,d} is defined over ℝ; from now on we will only consider its set V_{a,b,c,d}(ℝ) of ℝ-points. By Prop. III.1.1 of Morgan-Shalen [11] (p.458), V_{a,b,c,d}(ℝ) consists of characters corresponding to representations in either SU(2) or SL(2, ℝ). Since the leading term of the cubic polynomial defining V_{a,b,c,d} is xyz, its closure in ℝℙ^{3} meets the plane at infinity in three lines in general position (the singular cubic defined by xyz = 0 in ℝℙ^{2}). Thus for any values of (a, b, c, d) ∈ ℝ^{4}, the surface V_{a,b,c,d}(ℝ) has four ends. (Compare Figure 3.)

Figure 3. What V_{a,b,c,d}(ℝ) looks like near infinity

Theorem 4.1.1. Let (a, b, c, d) ∈ ℝ^{4}. Then V_{a,b,c,d}(ℝ) contains a singular point if and only if one of the following holds:

1. One of a, b, c, d equals ±2;

2. There is a reducible SU(2)-representation having boundary traces (a, b, c, d), and a, b, c, d ∈ [−2, 2];

3. There is a reducible SL(2, ℝ)-representation having boundary traces

(a, b, c, d), and a, b, c, d ∈ ℝ \ (−2, 2). The second condition is equivalent to

Δ(a, b, c, d) = 0 and a, b, c, d ∈ [−2, 2];

the third condition is equivalent to

Δ(a, b, c, d) = 0 and a, b, c, d ∈ ℝ \ (−2, 2).

We first verify the final statements. Suppose x ∈ ℝ satisfies

κ_{a,b}(x) = κ_{c,d}(x) = 0

and a, b, c, d ∈ [−2, 2]. By §3.4, Δ(a, b, c, d) = 0. Conversely, given Δ(a, b, c, d) = 0 with a, b, c, d ∈ [−2, 2], there exists a reducible SL(2, ℂ)- representation ρ with the given boundary traces. Furthermore,

x = tr(ρ(X)) ∈ [−2, 2]

by Lemma 2.3.1. By §2.2, the triples (a, b, x) and (c, d, x) correspond to SU(2)-representations. Amalgamating the representations as in the proof of Theorem 3.1.1 produces a SU(2)-representation ρ of π_{1}(M). The equivalence of the third condition is proved similarly.

The proof of the rest of Theorem 4.1.1 is analogous to the proof of Theorem 3.1.1. The only difference is that we must use Lemma 3.2.3 instead of Lemma 3.2.2. Thus, it suffices to show:

Lemma 4.1.2. Let x, a, b, c, d ∈ ℝ and define A_{1}(x), B_{1}(x), C_{1}(x) by (3.3.4). Then δ(x) ≥ 0 where δ(x) is defined in (3.2.5).

Proof. By (3.3.4),

A′_{1}(x) = 2x
B′_{1}(x) = −2(ac + bd)
C′_{1}(x) = −8x + 4(ab + cd)

and

δ(x) = B′_{1}(x)^{2} − 4A′_{1}(x)C′_{1}(x)
= 4(ac + bd)^{2} + 64x^{2} − 32x(ab + cd)
= 4 (4x − (ac + bd))^{2} ≥ 0.

4.2. Topological classification of the smooth real varieties. Assume that V_{a,b,c,d}(ℝ) is nonsingular. We now classify the possible topologies for V_{a,b,c,d}(ℝ). The six topological types are depicted in Figures 11, 12, 13, 14, 15, 16.

Theorem 4.2.1. If (a, b, c, d) ∈ ℝ^{4} and V_{a,b,c,d}(ℝ) is nonsingular, then V_{a,b,c,d}(ℝ) is diffeomorphic to one of the following:

1. A quadruply-punctured sphere, in the case a, b, c, d ∈ ℝ \ [−2, 2] with abcd < 0;

2. The disjoint union of a triply-punctured torus and a disk, in the case a, b, c, d ∈ ℝ \ [−2, 2] with abcd > 0;

3. The disjoint union of a triply-punctured sphere and a disk, in the case that one of a, b, c, d is in (−2, 2) and the rest are in ℝ\[−2, 2];

4. The disjoint union of an annulus and two disks, in the case that two of a, b, c, d are in (−2, 2) and two are in ℝ \ [−2, 2];

5. The disjoint union of four disks, in the case that three of a, b, c, d are in (−2, 2) and the other is in ℝ \ [−2, 2];

6. The disjoint union of four disks and a sphere, in the case that a, b, c, d ∈ (−2, 2)

In the second case, the component of V_{a,b,c,d} homeomorphic to a disc is the “Teichmüller space” of M, consisting of hyperbolic structures with geodesic boundary on M. Compare Goldman [6] and [5].

4.3. Reduction to a plane quartic curve. To prove Theorem 4.2.1, reduce the dimension by taking discriminants, as in the determination of the singularities. Again, difficulties arise because the discriminant f(x, y) is not monic in y. The set V_{a,b,c,d}(ℝ) contains two points for every (x, y) ∈ ℝ^{2} with f(x, y) > 0, one point for f(x, y) = 0, and none for f(x, y) < 0. Define

S = {(x, y) ∈ ℝ^{2} : f(x, y) > 0}

and

C = {(x, y) ∈ ℝ^{2} : f(x, y) = 0} = ∂S.

The projection

(x, y) : V_{a,b,c,d}(ℝ) ⟶ ℝ^{2}

expresses V_{a,b,c,d}(ℝ) as a union of two copies of S identified along their boundary C. This description as a “double branched cover” determines V_{a,b,c,d}(ℝ) up to homeomorphism (and hence diffeomorphism).

Fixing x ∈ ℝ with x ≠ ±2, for |y| large,

f(x, y) ∼ (x^{2} − 4)y^{2}

and so f(x, y) is positive if |x| > 2 and negative if |x| < 2. More precisely,

• for any x with |x| < 2, there exists M > 0 such that for |y| > M, (x, y) ∉ S;

• for any x with |x| > 2, there exists M > 0 such that for |y| > M, (x, y) ∈ S.

To study C, we reduce dimensions once more via discriminants. For fixed x ≠ ±2, the tangent of C at (x, y) for some y ∈ ℝ is vertical if and only if the discriminant

g(x) = 16κ_{a,b}(x)κ_{c,d}(x) = 0.

Lemma 4.3.1. Let x ∈ ℝ \ {±2}. Then the set of y ∈ ℝ such that (x, y) ∈ C consists of:

• Exactly two points if g(x) > 0;

• Exactly one point if g(x) = 0.

• No points if g(x) < 0.

(Recall the definition of A_{1}(x), B_{1}(x), C_{1}(x) from (3.3.4).)

Lemma 4.3.2. Suppose x = ±2 and g(x) ≠ 0. Then B_{1}(x) ≠ 0, and there exists exactly one y ∈ ℝ such that (x, y) ∈ C.

Lemma 4.3.3. Suppose that x = ±2 and g(x) = 0. Then B_{1}(x) = 0, and the set of y ∈ ℝ such that (x, y) ∈ C is empty, if C_{1}(x) ≠ 0 or ℝ, if C_{1}(x) = 0.

Proofs. We write f(x, y) as

f(x, y) = A_{1}(x)y^{2} + B_{1}(x)y + C_{1}(x)

Lemma 4.3.1 follows because f(x, y) is a quadratic polynomial in y with discriminant g(x). Lemma 4.3.2 follows because A_{1}(x) = 0 and

g(x) = B_{1}(x)^{2} − 4A_{1}(x)C_{1}(x).

Finally, Lemma 4.3.3 follows because its hypotheses imply f(x, y) = C_{1}(x).

Lemma 4.3.4. Suppose that a, b, c, d ∈ ℝ \ {2, −2}.

• If B_{1}(2) = C_{1}(2) = 0, then a = b and c = d.

• If B_{1}(−2) = C_{1}(−2) = 0, then a = −b and c = −d. Furthermore, in either of these cases, V_{a,b,c,d}(ℝ) is singular if and only if a, b, c, and d are either all in (−2, 2) or all in ℝ \ [−2, 2].

Proof. B_{1}(±2) = 0 factors as (b ∓ a)(c ∓ d) = 0, so without loss, a = ±b. Substituting, C_{1}(±2) = 0 becomes (b^{2} − 4)(c ∓ d)^{2} = 0. Since b ≠ ±2, we have c = ±d. One calculates Δ(±b, b, ±d, d) = 0. Now apply Theorem 4.1.1.

Lemma 4.3.5. Suppose V_{a,b,c,d}(ℝ) contains no singular points but g(x) has a real double root. Then this root is ±2, and a = ±b and c = ±d. One of the two possibilities must occur:

• a ∈ (−2, 2) and c ∈ ℝ \ [−2, 2];

• c ∈ (−2, 2) and a ∈ ℝ \ [−2, 2]. Furthermore C contains the line x = ±2.

Proof. By Theorem 4.1.1, this situation can only arise if g has a double root and yet some of a, b, c, d are in (−2, 2) and some are in ℝ \ [−2, 2]. Note that for κ_{a,b}(x) to have a double root, one of a or b must be ±2, but this does not happen since V_{a,b,c,d}(ℝ) is nonsingular. Thus, κ_{a,b}(x) and κ_{c,d}(x) have a common root. If this root lies in (−2, 2), then Lemma 2.3.1 implies that all of a, b, c, d lie in (−2, 2). Similarly, if this root lies in ℝ \ [−2, 2], then all of a, b, c, d lie in ℝ \ [−2, 2]. In either case, Theorem 4.1.1 implies that V_{a,b,c,d}(ℝ) is singular. Therefore, the root is ±2, implying a = ±b and c = ±d. In addition, B_{1}(±2) = C_{1}(±2) = 0, and so Lemma 4.3.3 implies that C contains the line x = ±2.

We continue the analysis of C. The solution of the quadratic equation f(x, y) = 0 for y when x ≠ ±2 is y = ϕ_{±}(x) where

ϕ_{±}(x) = \frac{−B_{1}(x) ± \sqrt{g(x)}}{2(x^{2} − 4)}

Note that B_{1}(x) is linear in x and g(x) ∼ 16x^{4}; so as |x| → ∞,

ϕ_{±}(x) → ±2.

Similarly, fix y ≠ ±2 and define x = ψ_{±}(y) by solving f(x, y) = 0. Then, as |y| → ∞,

ψ_{±}(y) → ±2.

Thus, for any ε > 0, there is M > 2 + ε for which

|x| > M ⟹ \begin{cases}|ϕ_{+}(x) − 2| & < ε \\ |ϕ_{−}(x) + 2| & < ε \end{cases}

and for which

|y| > M ⟹ \begin{cases}|ψ_{+}(y) − 2| & < ε \\ |ψ_{−}(y) + 2| & < ε \end{cases}

Fix 0 < ε < 1, and let Q be the square (−M, M)^{2}. Then C ∩ (ℝ^{2} \ Q) is homeomorphic to a disjoint union of eight copies of [M, ∞). For example, one of the eight homeomorphisms would map t ∈ [M, ∞) to (−t, ϕ_{+}(−t)). Note that C intersects ∂Q at the images of M under the eight homeomorphisms. C is smooth and hence a 1-manifold, so \overline{Q}∩C is a compact 1-manifold with boundary ∂\overline{Q} ∩ C consisting of eight points.

Lemma 4.3.6. Assume V_{a,b,c,d}(ℝ) is non-singular. With Q as above, \overline{Q} ∩ C has finitely many connected components. Each component is homeomorphic to either a closed interval [−1, 1], if at least one (and hence exactly two) points of the component lie in ∂Q, or S^{1} otherwise. Furthermore, there must be exactly four components homeomorphic to an interval.

C must look something like Figure 4. Outside Q, all points of C are close to the eight asymptotes. By choosing Q sufficiently large, C will be transverse to the four sides of ∂Q and C ∩∂Q is a set of eight points,

P = {p_{1}, p_{2}, p_{3}, p_{4}, p_{5}, p_{6}, p_{7}, p_{8}},

numbered as in Figure 4. To determine the topologies of C and S, it suffices to decide how these eight points of C ∩ ∂Q are joined pairwise inside Q, and to find the number and positions of any bounded components.

Figure 4. How the curve C meets the square Q

In each of the following lemmas, the set of ℝ-points is assumed to be nonsingular.

Lemma 4.3.7. If p_{n} is joined by C to p_{m}, then n + m is odd.

Proof. Suppose n+m is even. Then on ∂Q, the points p_{n} and p_{m} separate the remaining points p_{i} into two sets P_{1} and P_{2} of odd cardinality. Since an arc α of C ∩ Q joins p_{n} and p_{m}, this arc separates P_{1} from P_{2} in Q̄. Furthermore the points of

P − {p_{m}, p_{n}} = P_{1} ∐ P_{2}

are pairwise joined by arcs of C ∩ Q which (since C is smooth) do not meet α. Thus the points of P_{1} join to points of P_{1}, and similarly for P_{2}. This partitions P_{1} into disjoint pairs, contradicting the odd cardinality of P_{1}.

Lemma 4.3.8.

• p_{3} joins to p_{4} if and only if g(x) has a root in (2, ∞);

• p_{7} joins to p_{8} if and only if g(x) has a root in (−∞, −2).

Proof. We’ll show the first statement; the second is similar. Suppose first that p_{3} joins to p_{4}. The component joining p_{3} and p_{4} must have a vertical tangent — corresponding to a root of g(x) — at some point x_{0}. If x_{0} < 2, then the component crosses the line x = 2 twice, contradicting Lemma 4.3.2. (If the component crossed the line at the same point each time, there would be a singularity. Alternatively, if g(2) = 0, then by Lemma 4.3.3, C must contain the whole line x = 2, and we would have singularities where the component crosses.) If x_{0} = 2, then, by Lemma 4.3.3, either C doesn’t intersect the line x = 2, or else C contains it. But a vertical tangent at x = 2 would force at least one point in C on that line; and if C contained the line, a component from the right touching the line at a point would result in a singularity. Thus x_{0} > 2. Since C has a vertical tangent there, g(x_{0}) = 0.

Conversely, let x_{0} > 2 be the root in question. Without loss, x_{0} is the largest root of g. Thus g(s) < 0 for some s ∈ (2, x), since g(x) > 0 for x > x_{0}, and x_{0} is not a double root. If p_{3} were to join to p_{1}, p_{6}, p_{7}, or p_{8}, then clearly C would cross the line x = s, contradicting Lemma 4.3.1. It also can’t join to p_{2} or p_{5}, since in that case, C would cross x = s somewhere along the approach to the asymptote at x = 2. Thus p_{3} must join to p_{4}.

Lemma 4.3.9.

• p_{1} joins to p_{6} if and only if g has two roots in (−∞, −2] and two roots in [−2, 2];

• p_{2} joins to p_{5} if and only if g has two roots in [2, ∞) and two roots in [−2, 2].

Proof. We’ll show the first statement; the second is similar. To prove the forward direction, suppose p_{1} and p_{6} are joined by a component. If this component is the line x = −2, then Lemmas 4.3.3 and 4.3.4 imply that a = −b and c = −d. Thus g has a double root at −2. By Lemma 4.3.5, we can assume without loss that a, b ∈ (−2, 2) and c, d ∈ ℝ\[−2, 2]. Therefore, by Lemma 2.3.1, g has a root in (−∞, −2], a double root at −2, and a root in (−2, 2], and we are done.

Now assume C does not contain the line x = −2. Consider the whole component of C that joins p_{1} to p_{6}, and not just the part lying inside Q; let α be a diffeomorphism mapping ℝ onto this component. If −2 is a root of g, Lemma 4.3.3 implies that C does not intersect the line x = −2. If the x-value of α(t) increases from −2 as t increases from −∞, then at some point it must decrease back to −2 as t increases to +∞. Thus, there must be a vertical tangent — and hence a root of g — at some x-value greater than −2. This root must be less than 2, for the same reasons as in Lemma 4.3.8. Meanwhile, p_{7} and p_{8} must join to each other, since they have nowhere else to go. Lemma 4.3.8 implies that g has a root in (−∞, −2). We now know that g has at least three real roots; it therefore has a fourth. By Lemma 2.3.1, this fourth root is somewhere in (−∞, 2], and we have the desired conclusion. The argument is similar if the x-value of α decreases and then increases.

On the other hand, if −2 is not a root of g, Lemma 4.3.1 implies that C must cross the line x = −2 at exactly one point. If any component other than the one in question were to cross this line, then it must cross it at least twice (once to get across, and once to get back) or at a singularlity or vertical tangent. Thus, the component in question must cross x = −2 exactly once. The x-value of α must increase from −2, decrease back down to −2 and below, and then increase back up to −2; as a result, we get at least two vertical tangents, one on each side of −2. By the same argument as above, both roots must be less than 2. Lemma 2.3.1 implies g must also have two other roots, one on each side of −2, and neither larger than 2.

For the converse, suppose that −2 is a simple root of g, and x_{1} ∈ (−∞, −2) and x_{2}, x_{3} ∈ (−2, 2] are also roots. Lemma 4.3.8 implies that p_{7} and p_{8} join. Furthermore, g is negative on (x_{2}, x_{3}), and so there are no points of C above that interval. Thus p_{1} and p_{6} are cut off from the other points, and must join to each other. The argument is similar if −2 is a root, with two roots less than −2 and one greater, or if there are two roots less than −2 and two greater. Finally, if −2 is a multiple root of g, then Lemma 4.3.5 implies a = −b and c = −d. As a result, B_{1}(−2) = C_{1}(−2) = 0. Lemma 4.3.3 implies that C contains the line x = −2, and so p_{1} and p_{6} join.

(Note that in the above lemma, the case of a double root at ±2 is considered as a deformation of the case of one root just above and one just below.)

Lemma 4.3.10. If C has a compact component, then g has four distinct real roots x_{1} < x_{2} < x_{3} < x_{4}, and the projection of the compact component onto the x-axis is [x_{2}, x_{3}].

Proof. The projection of the component onto the x-axis must be [x_{i}, x_{j}], where x_{i} and x_{j} are distinct roots of g. Furthermore, g must be strictly positive on (x_{i}, x_{j}), and negative for some x < x_{i} and for some x > x_{j}. Thus, g must have four real roots, and x_{i} and x_{j} must be the middle two.

Corollary 4.3.11. There is at most one compact component of C.

Lemma 4.3.12. If g has four real roots x_{1} < x_{2} < x_{3} < x_{4} and 2, −2 ∉ [x_{2}, x_{3}], then C has a compact component projecting onto [x_{2}, x_{3}].

Proof. Since the leading coefficient of f is nonzero throughout [x_{2}, x_{3}], and the discriminant g is zero at x_{2} and x_{3} and positive in between, the part of C that lies above the [x_{2}, x_{3}] must consist of two line segments glued at the endpoints. As g is negative just below x_{2} and just above x_{3}, it follows C has a compact component.

4.4. The six cases. We now analyze individual cases to find the resulting topologies.

In the first case, we are given a, b, c, d ∈ ℝ \ [−2, 2] and abcd < 0. By applying Lemma 2.7.1, we can assume that a, b, c > 2 and d < −2. Thus, by Lemma 2.3.1, g has two roots in [2, +∞) and two in (−∞, 2]. Note that by Lemma 4.3.5, the roots of g are all distinct; in particular, there must be at least one root greater than 2. Now by Lemma 4.3.8, p_{3} and p_{4} join, and p_{7} and p_{8} join. p_{1} and p_{5} do not join, by Lemma 4.3.7. If p_{1} and p_{6} were to join, then by Lemma 4.3.9, all four roots would be in (−∞, 2], contradicting the existence of a root greater than 2. Thus, p_{1} and p_{2} join; finally, p_{5} and p_{6} must join. Furthermore, there cannot be a compact component, since by Lemma 4.3.10 such a component would have to project onto [x_{2}, x_{3}], with x_{2} ≤ −2 and x_{3} ≥ 2; but such a component would cross the lines x = ±2 twice, contradicting Lemma 4.3.2 and Lemma 4.3.3. Thus, C ∩ \overline{Q} consists of just the four above-mentioned components.

Since f is negative for x = 0 and y large, we see that S is homeomorphic to a closed disk minus four points of the boundary. (See Figure 5.) Doubling along the boundary, we see that V_{a,b,c,d}(ℝ) is homeomorphic, and hence diffeomorphic, to a quadruply-punctured sphere. (See Figure 11.)

In the second case, all a, b, c, d ∈ ℝ \ [−2, 2] and abcd > 0. Applying Lemma 2.7.1, we may assume that a, b, c, d > 2. Lemma 2.3.1 implies that g(x) has four roots in [2, +∞). Lemma 4.3.8 implies that p_{3} and p_{4} must join, and p_{7} and p_{8} cannot join. Lemma 4.3.9 implies that p_{1} and p_{6} cannot join, nor can p_{2} and p_{5}. Thus p_{1} joins either p_{2} or p_{8}, by Lemma 4.3.7. If p_{1} joins p_{2}, then p_{7} must join p_{6} and so p_{8} joins p_{5}; or, if p_{1} joins p_{8}, then p_{5} joins p_{6} and p_{2} joins p_{7}. Meanwhile, Lemma 4.3.12 implies that C has a compact component somewhere between the component joining p_{3} to p_{4} and the line x = 2. Thus C consists of a circle, together with one of the two possible sets of four non-compact components; both setups result in the same topology for S.

Since f is negative for x = 0 and y large, S is homeomorphic to a closed annulus minus three points of the outer boundary, along with a closed disk minus one point of the boundary. (See Figure 6.) Doubling

Figure 5. Case 1

again, V_{a,b,c,d}(ℝ) is homeomorphic, and hence diffeomorphic, to a triplypunctured torus and a disk. (See Figure 12.)

In the third case, three of a, b, c, d ∈ ℝ \ [−2, 2] and the fourth in (−2, 2). By applying Lemma 2.7.1, we can assume that d ∈ (−2, 2) and a, b, c > 2. By Lemma 2.3.1, g has two roots in [2, +∞) and no other real roots. Thus, by Lemma 4.3.10, there cannot be a compact component. Furthermore, Lemma 4.3.8 and Lemma 4.3.9 imply that p_{3} and p_{4} must join, but p_{7} and p_{8} cannot join, p_{1} and p_{6} cannot join, and p_{2} and p_{5} cannot join. Thus, we end up with the same arrangement of non-compact components as in Case 2, but no compact component.

From the sign of f, we see that S is homeomorphic to a closed disk minus three points of the boundary, along with a closed disk minus one point of the boundary. (See Figure 7.) Doubling again, we see that V_{a,b,c,d}(ℝ) is homeomorphic, and hence diffeomorphic, to a triplypunctured sphere and a disk. (See Figure 13.)

Figure 6. Case 2

In the fourth case, two of a, b, c, d ∈ ℝ \ [−2, 2] and two in (−2, 2). Applying Lemma 2.7.1, we can assume that a, b ∈ (−2, 2) and c, d > 2. There are two possibilities.

Suppose first that g(x) has a double root. Lemma 4.3.5 implies this double root must be 2; there is also one root in [−2, 2) and one in (2, ∞). By Lemma 4.3.10, there is no compact component and Lemma 4.3.5 implies that p_{2} and p_{5} join by a vertical line. Lemma 4.3.8 implies that p_{3} and p_{4} join, but p_{7} and p_{8} do not. Thus p_{1} and p_{8} must join, as must p_{6} and p_{7}.

The second possibility is that all of the roots of g are distinct; thus, two distinct roots lie in [−2, 2] and two distinct roots lie in [2, +∞), and 2 is not a double root. Once again, Lemma 4.3.10 precludes a compact component, and Lemma 4.3.8 implies that p_{3} and p_{4} join. In addition, Lemma 4.3.9 implies that p_{2} and p_{5} join, while p_{1} and p_{6} do not. Thus, p_{1} and p_{8} join, as do p_{6} and p_{7}.

Both possibilities give the same result: S is homeomorphic to a closed disk minus two points of the boundary, along with two closed

Figure 7. Case 3

disks missing one boundary point each. (See Figure 8.) Doubling again, V_{a,b,c,d}(ℝ) is homeomorphic, and hence diffeomorphic, to a disjoint union of an annulus and two disks. (See Figure 14.)

In the fifth case, one of a, b, c, d lie in ℝ \ [−2, 2] and the other three in (−2, 2). Applying Lemma 2.7.1, we can assume that a, b, c ∈ (−2, 2) and d > 2. Lemma 2.3.1 implies g has two roots in [−2, 2] and no other real roots. Lemma 4.3.10 precludes a compact component. Furthermore, Lemma 4.3.8 and Lemma 4.3.9 imply that p_{3} can’t join to p_{4}, p_{7} can’t join to p_{8}, p_{1} can’t join to p_{6}, and p_{2} can’t join to p_{5}. Furthermore, g is negative for some x_{0} ∈ (−2, 2); thus, none of the points on the left can join to any of the points on the right, by Lemma 4.3.1. Therefore p_{1} joins to p_{8}, p_{2} joins to p_{3}, p_{4} joins to p_{5}, and p_{6} joins to p_{7}.

From the sign of f, we see that S is homeomorphic to a union of four disjoint closed disks missing one boundary point each. (See Figure 9.) Doubling, we see that V_{a,b,c,d}(ℝ) is homeomorphic, and hence diffeomorphic, to a union of four disjoint disks. (See Figure 15.)

Figure 8. Case 4

In the sixth case, all of a, b, c, d lie in (−2, 2). Lemma 2.3.1 implies that g has four roots in [−2, 2]. Lemma 4.3.12 implies that C has a compact component between the lines x = −2 and x = 2. The rest of the analysis is exactly as in Case 5; the result is that p_{1} joins to p_{8}, p_{2} joins to p_{3}, p_{4} joins to p_{5}, and p_{6} joins to p_{7}.

From the sign of f, we see that S is homeomorphic to a closed disk, along with four closed disks missing one boundary point each. (See Figure 10.) Doubling, we see that V_{a,b,c,d}(ℝ) is homeomorphic, and hence diffeomorphic, to four disks and a sphere. (See Figure 16.) This concludes our case-by-case analysis and the proof of Theorem 4.2.1.

5. A compact component of SL(2, ℝ)-representations

When (a, b, c, d) ∈ ℝ and V_{a,b,c,d}(ℝ) is smooth, then each connected component of V_{a,b,c,d}(ℝ) consists of characters of representations in either SL(2, ℝ) or SU(2). Components of spaces of representations in SU(2) are compact. Conversely, when M is a once-punctured torus,

Figure 9. Case 5

the only compact components of 𝔛(M)(ℝ)(t) are characters of SU(2)- representations. (See Goldman [6] for a description of the topology of the relative character varieties for a punctured torus.) However, for the quadruply punctured sphere, there are compact components of V_{a,b,c,d}(ℝ) for certain values of (a, b, c, d) which do not correspond to SU(2)-representations.

If V_{a,b,c,d}(ℝ) contains a compact component, then Theorem 4.2.1 implies that

−2 < a, b, c, d < 2,

and conversely, every such (a, b, c, d) ∈ (−2, 2)^{4} for which V_{a,b,c,d}(ℝ) is smooth contains a compact component. (Compare the analysis of Case 6.) Here is a criterion for when this compact component consists of SL(2, ℝ)-characters or SU(2)-characters, in terms of the function Δ(a, b, c, d) defined in §3.4.

Proposition 5.1. Suppose a, b, c, d ∈ (−2, 2) and V_{a,b,c,d} is smooth. Then the compact component of V_{a,b,c,d}(ℝ) consists of characters of

Figure 10. Case 6

SL(2, ℝ)-representations if and only if

(5.0.1)  Δ(a, b, c, d) > 0
(5.0.2)  16 − abcd − 2(a^{2} + b^{2} + c^{2} + d^{2}) > 0.

Otherwise, each element in the component is the character of an SU(2)- representation.

Proof. Decompose the quadruply-punctured sphere M along ξ, as in §2.5, as a union of two triply-punctured spheres. A given representation π_{1}(M) ⟶ SL(2, ℂ) has image in SU(2) only if its restriction to the fundamental group of each component of M|ξ has image in SU(2). Furthermore there is an SU(2) character on M with boundary traces (a, b, c, d) if and only if there is an x such that (a, b, x) and (c, d, x) are SU(2) characters on triply-punctured spheres. To construct this character, amalgamate the two representations along the common boundary X as in §2.5; since two elements of SU(2) have the same trace if and only if they are conjugate, we can ensure that the two representations agree on this boundary.

Recall from §2.2 that (a, b, x) is an SU(2)-character for the triplypunctured sphere if and only if a, b, x ∈ [−2, 2] and κ_{a,b}(x) ≤ 0. For a, b ∈ ℝ, define the interval I_{a,b} ⊂ ℝ by:

I_{a,b} = {x ∈ ℝ | κ_{a,b}(x) ≤ 0}
= [i_{a,b}^{−}, i_{a,b}^{+}]

where

i_{a,b}^{−} = \frac{ab − \sqrt{(a^{2} − 4)(b^{2} − 4)}}{2}
i_{a,b}^{+} = \frac{ab + \sqrt{(a^{2} − 4)(b^{2} − 4)}}{2}

Since κ_{a,b}(±2) ≥ 0,

I_{a,b} ⊂ [−2, 2]

For given (a, b, c, d) with −2 < a, b, c, d < 2, the values of x ∈ ℝ such that (a, b, c, d, x) extends to a character of an SU(2)-representation comprise the intersection

I_{a,b} ∩ I_{c,d} ⊂ [−2, 2].

Furthermore the possible (y, z) ∈ ℝ^{2} comprise a conic, which by (2.6.2), is an ellipse. Thus characters of SU(2)-representations with boundary traces (a, b, c, d) form a pencil of ellipses over the interval I_{a,b} ∩ I_{c,d}.

However, if I_{a,b} ∩ I_{c,d} = ∅, there will be a compact component of V_{a,b,c,d}(ℝ) not corresponding to SU(2)-representations. By applying a permutation if necessary, assume that

i_{a,b}^{+} < i_{c,d}^{−}

Then a component of V_{a,b,c,d}(ℝ) will map under

x : V_{a,b,c,d}(ℝ) ⟶ ℝ

to the interval [i_{a,b}^{+}, i_{c,d}^{−}]. By (2.6.2), the restriction of x to x^{−1}([−2, 2]) is proper. If

x_{0} ∈ I_{a,b} ∪ I_{c,d}

then

\frac{κ_{a,b}(x_{0})κ_{c,d}(x_{0})}{4 − x_{0}^{2}} < 0

and by (2.6.2), the preimage x^{−1}(x_{0}) = ∅. Thus

x^{−1}([i_{a,b}^{+}, i_{c,d}^{−}])

is a compact component of V_{a,b,c,d}(ℝ) not containing characters of SU(2)-representations. Since real characters are characters of either SU(2)-representations or SL(2, ℝ)-representations, this compact component must consist of SL(2, ℝ)-representations.

Thus, the compact component of V_{a,b,c,d}(ℝ) consists of SU(2)-repre-sentations if and only if

I_{a,b} ∩ I_{c,d} ≠ ∅

and otherwise consists of SL(2, ℝ)-representations. The latter case occurs when

i_{a,b}^{+} < i_{c,d}^{−}

or

i_{c,d}^{+} < i_{a,b}^{−}.

These conditions are equivalent to conditions (5.0.1) and (5.0.2) by the following argument:

The condition i_{a,b}^{+} < i_{c,d}^{−} is equivalent to:

\sqrt{(a^{2} − 4)(b^{2} − 4)} + \sqrt{(c^{2} − 4)(d^{2} − 4)} < cd − ab

and i_{c,d}^{+} < i_{a,b}^{−} is equivalent to:

\sqrt{(a^{2} − 4)(b^{2} − 4)} + \sqrt{(c^{2} − 4)(d^{2} − 4)} < ab − cd

The following lemma (whose straightforward proof is omitted) then applies:

Lemma 5.0.1. Let p, q, r be nonnegative real numbers. Then

\sqrt{p} + \sqrt{q} < r

if and only if

r^{4} − 2r^{2}(p + q) + (p − q)^{2} > 0
r^{2} − p − q > 0.

Taking

p = (a^{2} − 4)(b^{2} − 4)
q = (c^{2} − 4)(d^{2} − 4)
r = |ab − cd|

and calculating that

r^{4} − 2r^{2}(p + q) + (p − q)^{2} = 16Δ(a, b, c, d)
r^{2} − p − q = 2(16 − abcd − 2(a^{2} + b^{2} + c^{2} + d^{2})),

the compact component corresponds to SL(2, ℝ)-representations if and only if (5.0.1) and (5.0.2).

Here is an explicit example. Let a, b, c = 3/2 and d = −3/2. For example,

α = [ \begin{matrix}1 & 1 \\ −1/2 & 1/2 \end{matrix} ] , β = [ \begin{matrix}1 & 1/2 \\ −1 & 1/2 \end{matrix} ] , γ = [ \begin{matrix}1 & 1 \\ −1/2 & 1/2 \end{matrix} ]

generates an SL(2, ℝ)-representation with the desired character. Furthermore, its (x, y, z) traces are (0, 0, 1/4). Now the equation for the relative character variety at these boundary traces is

x^{2} + y^{2} + z^{2} + xyz − 1/16 = 0

which does not intersect the planes x = ±2, y = ±2, or z = ±2 at real points.

References

1. Benedetto,R., Senior Thesis, Harvard University (1993)

2. Benedetto,R. and Goldman, W., Investigating moduli spaces with Geomview, in “Imagine That!,” A quarterly publication of the Geometry Center, (1994)

3. Fricke, R., and Klein, F., Vorlesungen der Automorphen Funktionen, Teubner, Leipzig, Vol. I (1897), VolİI (1912)

4. Geomview, Software from the Geometry Center, University of Minnesota

5. Goldman, W., Geometric structures and varieties of representations, in “The Geometry of Group Representations,” Contemp. Math. 74, Amer. Math. Soc. (1988), 169–198

6. Goldman, W., Topological components of spaces of representations, Inv. Math. Vol. 93 No. 3, (1988), pp. 557–607.

7. Goldman, W., Ergodic theory on moduli spaces, Ann. Math. (to appear)

8. Lubotzky, A. and Magid, A., Varieties of representations of finitely generated groups,, Memoirs Amer. Math. Soc. 336 (vol. 58) (1985)

9. Magnus, W., Rings of Fricke characters and automorphism groups of free groups, Math. Zeit. 170 (1980), 91–103

10. Mathematica, Software from Wolfram Research, Inc.

11. Morgan, J.W. and Shalen, P.B., Valuations, trees and degenerations of hyperbolic structures I, Ann. Math. 120 (1984), 401–476

12. Procesi, C., Invariant theory of n by n matrices, Adv. Math. 19 (1976), 306–381

List of Figures

1 A triply-punctured sphere and its three boundary components 7 2 A quadruply-punctured sphere 9 3 What V_{a,b,c,d}(ℝ) looks like near infinity 18 4 How the curve C meets the square Q 23 5 Case 1 27 6 Case 2 28

RELATIVE CHARACTER VARIETIES 37

7 Case 3 29 8 Case 4 30 9 Case 5 31 10 Case 6 32 11 Class 1: abcd < 0 and none of |a|, |b|, |c|, |d| < 2 37 12 Class 2: abcd > 0 and none of |a|, |b|, |c|, |d| < 2 37 13 Class 3: One of |a|, |b|, |c|, |d| < 2 38 14 Class 4: Two of |a|, |b|, |c|, |d| < 2 38 15 Class 5: Three of |a|, |b|, |c|, |d| < 2 39 16 Class 6: All of |a|, |b|, |c|, |d| < 2 39

Mathematics Department, Brown University, Providence, RI 02906 USA (Benedetto), Mathematics Department, University of Maryland, College Park, MD 20742 USA (Goldman)

E-mail address: benny@math.brown.edu (Benedetto),

wmg@math.umd.edu (Goldman)

Figure 11. Class 1: abcd < 0 and none of |a|, |b|, |c|, |d| < 2

Figure 12. Class 2: abcd > 0 and none of |a|, |b|, |c|, |d| < 2 Figure 13. Class 3: One of |a|, |b|, |c|, |d| < 2

Figure 14. Class 4: Two of |a|, |b|, |c|, |d| < 2 Figure 15. Class 5: Three of |a|, |b|, |c|, |d| < 2

Figure 16. Class 6: All of |a|, |b|, |c|, |d| < 2