Spaces of Surface Group Representations William Goldman University - - PDF document

Spaces of Surface Group Representations

William Goldman University of Maryland Geometry Conference honouring

Nigel Hitchin

Consejo Superior de Investigaciones Cientficas, Madrid 5 September 2006

Historical context of surface group representations, and some recent developments arising from two sem- inal papers of Nigel Hitchin: The self-duality equations on Riemann surfaces,

• Proc. London Math. Soc. 55 (1987), 59–126.

Lie groups and Teichm¨ uller space, Topology 31 (3), (1992), 449–473.

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When π is the fundamental group of a closed surface S, and G is a Lie group, the set Hom(π, G)/G

• f equivalence classes of homomorphisms

π − → G is a natural geometric object. When G is an R-algebraic group, it is an R-algebraic set.

• How many connected components does

it have?

• What are the dimensions of these com-

ponents?

• Describe the singularities of these com-

poenents.

• Determine the homotopy type of these

components.

• What geometric structures do these com-

ponents enjoy?

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The theory of Higgs bundles, pioneered by Hitchin and Simpson, sheds light by impos- ing a conformal structure on S, that is, considering a Riemann surface Σ ≈ S. Hom(π, G)/G admits a natural action of the mapping class group π0(Diff(S) ∼ = Out(π) := Aut(π)/Inn(π); Which structures are independent of Σ (and hence Out(π)-invariant)? Primary examples:

• G = PSL(2, R), orientation-preserving

isometries of H2;

• G = PSL(2, C), orientation-preserving

isometries of H3;

• G = SU(n, 1), holomorphic isometries
• f Hn

C;

• G = PSL(n + 1, R), collineations of

RPn;

• G = Sp(4) ∼ SO(3, 2).

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Hyperbolic structures on surfaces The following constructions are equivalent:

• Curvature −1 Riemannian metric on S;
• Alas of H2-valued coordinate charts on

S, locally isometric coordinate changes;

• A pair (ρ, dev), representation

π

ρ

− → PSL(2, R), equivariant immersion ˜ S

dev

− → H2;

• (ρ, dev) with dev isometry;
• (ρ, dev) with ρ Fuchsian

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Fuchsian representations of surface groups ρ is Fuchsian :

• Embedding with discrete image;
• Free proper isometric π-action on H2;
• ρ(γ) is hyperbolic ∀γ = 1.
• hyperbolic projective actions on RP1.

Fuchsian representations comprise connected components of Hom(π, PSL(2, R)) (Weil 1960). Uniformization theorem establishes equiv- alence between (equivalence classes of): Fuchsian reps π − → PSL(2, R)

• ←

→ Marked conformal structures Σ ≈ S

• which is the Teichm¨

uller space TS of S.

6

Characteristic classes Euler class of associated H2-bundle defines: Hom(π1(S), G)

Euler

− − → H2(S; Z) ∼ = Z satisfying |Euler(ρ)| ≤ |χ(S)| = 2g − 2. (Milnor 1958, Wood 1971) Generalization: X is Hermitian symmetric space of noncom- pact type, G = Aut(X), integrating (nor- malized) K¨ ahler form over a section of the flat X-bundle with holonomy π

ρ

− → G = ⇒ Hom(π, G)

τ

− → Z satisfying |τ(ρ)| ≤ (2g − 2)rankR(G). (Domic-Toledo 1987) The invariant is a bounded cohomology class in H2(G) ρ is maximal :⇐ ⇒ |τ(ρ)| = (2g − 2)rankR(G).

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PSL(2, R)-representations Theorem (Goldman 1980). ρ is maximal

• ρ is Fuchsian.

Theorem (Goldman 1987, Hitchin 1986). The connected components of Hom(π, PSL(2, R)) are the 4g − 3 preimages Re := {ρ | Euler(ρ) = e} for e = 2 − 2g, 3 − 2g, . . . , 0, . . . , 2g − 2. Maximal representations thus correspond to hyperbolic structures on S. What do representations in other compo- nent correspond to?

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Some examples

• Take a regular octagon with angles π/4.

Pair the sides by a1, b1, a2, b2 ∈ PSL(2, R) according to the pattern below. Around the single vertex in the quotient is a cone of angle 8(π/4) = 2π, a disc in the hyperbolic plane. These isometries satisfying the defining rela- tion for π1(S): a1b1a−1

1 b−1 1 a2b2a−1 2 b−1 2

= 1 and therefore defines a representation π1(S)

ρ

− → PSL(2, R). The identification space is a hyperbolic surface with g = 2 with Fuchsian holo- nomy representation of Euler class −2 = χ(S).

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• Take a regular right-angled
• ctagon.

Again, side pairings a1, b1, a2, b2 exist. Now 8 right angles compose a neigh- borhood of the vertex in the quotient space, and the quotient space is a hy- perbolic structure with one singularity

• f cone angle

4π = 8(π/2). Since the product a1b1a−1

1 b−1 1 a2b2a−1 2 b−1 2

is rotation through 4π (the identity), the holonomy representation ˆ ρ of the nonsingular hyperbolic surface S \ {p} extends: π1

• S \ {p}
• ˆ

ρ

• π1(S)

ρ

• PSL(2, R)

and Euler(ρ) = −1. Although ρ(π) is discrete, ρ is not injective.

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• Take a hyperbolic surface M with ho-

lonomy π1(M)

ψ

− → PSL(2, R) and a degree-one map S

f

− → M not homotopic to a homeomorphism. (For example, collapse ≥ 1 handles). The composition π = π1(S)

f∗

− → π1(M)

ψ

֒ → PSL(2, R) is a (discrete, nonfaithful) representa- tion ρ of Euler class χ(M) > χ(S) which is not the holonomy of a branched structure. (Branched structures with holonomy ρ ← → branched coverings S − → M homotopic to f. But deg(f) = 1 = ⇒ f must be a homeomorphism.)

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Higgs bundles An GL(n, C)-Higgs pair over a Riemann surface Σ consists of a holomorphic vector bundle V over Σ together with a holomor- phic End(V )-valued 1-form Φ. The Higgs pair (V, Φ) is stable ⇐ ⇒ ∀ Φ-invariant holo- morphic subbundles W ⊂ V , deg(W) rank(W) < deg(V ) rank(V ). Theorem (Hitchin-Donaldson-Corlette-Simpson). There is a natural bijection between equiv- alences classes: Stable Higgs pairs (V, Φ) over Σ

• ←

→

• Irreducible rep’ns

π1(Σ)

ρ

− → GL(n, C)

• Φ = 0: Stable holomorphic vector bundles

← → Irreducible U(n)-representations. (Narasimhan-Sesadri) New ingredient, when G is noncompact: Existence/uniqueness of harmonic mappings

• f compact Riemannian manifolds.

(Eels-Sampson)

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Harmonic metrics Going from ρ to (V, Φ) involves finding a harmonic metric, Hermitian metric h on the flat Cn-bundle Eρ with holonomy ρ, whose corresponding ρ-equivariant map

• M
• h

− → GL(n, C)/U(n) is harmonic map into the symmetric space. Harmonic metric h determines Higgs pair:

• Higgs field Φ ←

→ ∂h ∈ Ω1 End(V )

• .
• Holomorphic structure ¯

∂V on V arises from conformal structure Σ and the unique Hermitian connection ∇h preserving h. The self-duality equations: (dA)′′(Φ) = 0 F(A) + [Φ, Φ∗] = 0 A is h-unitary connection, F(A) curvature, Φ∗ adjoint of Φ with respect to h.

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PSL(2, R)-representations and Higgs bundles Choose 2g − 2 ≤ e < 0. Hitchin identifies component Re with Euler class e with Higgs pairs (V, Φ) where V = L1 ⊕ L2 direct sum of line bundles deg(L1) − deg(L2) = e. Let D ≥ 0 be an effective divisor deg(D) = d < 2g − 2. L1 = κ−1/2 ⊗ D L2 = κ1/2.

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Then (V, Φ) stable Higgs pair with Higgs field corresponding to Φ =

• 0 sD

Q 0

• .

where:

• sD meromorphic section of D, giving

Higgs field in κ ⊗ Hom(L2, L1) ∼ = D ⊂ Ω1 Σ, End(V )

• ;
• Q ∈ H0(Σ, κ2) holomorphic quadratic

differential with div(Q) ≥ D. determining Higgs field in κ ⊗ Hom(L1, L2) ∼ = κ2 ⊂ Ω1 Σ, End(V )

• ;

When Q = 0, this Higgs bundle corresponds to the uniformization representation. In general, when d = 0, the harmonic metric is a diffeomorphism (Schoen-Yau 1979) and Q is its Hopf differential.

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Symmetric products The Euler class e relates to the degree d by: e = 2 − 2g + d Theorem (Hitchin 1987). Re identifies with a holomorphic vector bundle over the sym- metric power Symd(Σ). The fiber over D ∈ Symd(Σ) is the 3g−3−d-dimensional vector space {Q ∈ H0(Σ, κ2) | div(Q) ≥ D}. Q ← → Hopf differential of h. When e = 2−2g, then d = 0 and the space (TS) of Fuchsian representations identifies with the vector space H0(Σ, κ2). The zero section ← → Higgs pairs where the harmonic metric h is holomorphic: Branched hyperbolic structures, — singular hyperbolic structures with coni- cal singularities with angles multiples of 2π.

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Uniformization with singularities Branched conformal structures on S with cone angles θ1, . . . , θk ∈ 2πZ+ form bundle Sd of symmetric powers Symd(Σ)

• ver TS where

d = 1 2π

k

• i=1

(θi − 2π). Theorem (Hitchin, McOwen, Troyanov). Given branched conformal structure (Σ, D) with 2 − 2g +

k

• i=1

(θi − 2π) > 0, ∃ ! branched hyperbolic structure confor- mal to (Σ, D). Resulting uniformization map Sd U − → H2−2g+d ⊂ Hom(π, G)/G is homotopy equivalence. (Hitchin) But it’s not surjective!

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Quasi-Fuchsian representations When the representation is deformed π − → PSL(2, R) ֒ → PSL(2, C) the action on CP1 is topologically conjugate to the original Fuchsian action. ∃ H¨

• lder ρ-equivariant embedding

S1 − → CP1, whose image has Hausdorff dimension > 1, — unless the deformation is still Fuchsian. The space of such representations is quasi- Fuchsian space QFS.

• QFS ←

→ TS×TS. (Ahlfors-Bers 1960)

• QFS consists exactly of the discrete

embeddings. (Marden’s tameness conjecture, estab- lished 2004 by Agol, Calegari—Gabai, Choi, following Thurston 1979, Bona- hon 1986)

• ∂QFS is nonrectifiable, and is near non-

discrete representations.

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Complex hyperbolic structures In contrast, Fuchsian representations in PU(n, 1) π − → SL(2, R) ∼ = SU(1, 1) ֒ → PU(n, 1), are locally rigid (Goldman 1983), and glob- ally rigid (Toledo 1989): Theorem (Toledo). π

ρ

− → PU(n, 1) is maximal ⇐ ⇒ ρ is a discrete embedding preserving a complex geodesic, that is ρ(π) ⊂ U(1, 1) × U(n − 1). In particular components

• f

maximal representations has dimension smaller than the expected dimension! Thus connected components (in the classi- cal topology) exist which consist entirely of reducible representations.

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Higher rank Hermitian spaces Using infinite-dimensional Morse theory on Higgs bundles, Bradlow, Garc´ ıa-Prada and Gothen began a detailed analysis into the topology of the deformation spaces. In particular they determined the connected components (and much of the topology) of representation spaces Hom(π, G)/G where G = U(p, q) in terms of discrete invariants. Theorem (B-G-G 2003). Hom

• π, U(p, q)
• has 2(p + q) min(p, q) (g − 1) + gcd(p, q)

connected components.

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Maximal representations In this case rankR(G) = p ≤ q. Suppose that ρ is maximal, that is, |τ(ρ)| = p(2g − 2). In their investigation, Bradlow, Garc´ ıa-Prada, and Gothen found a surprising property of maximal representations: Every such representation conjugate to U(p, p) × U(q − p). Theorem (Burger–Iozzi–Wienhard 2003). Let X be a Hermitian symmetric space, and maximal representation π

ρ

− → G.

• Zariski closure L of ρ(π) is reductive;
• Symmetric space associatied to L is

a Hermitian symmetric tube domain;

• ρ is a discrete embedding.

Conversely: X is a tube domain, = ⇒ ∃ maximal ρ with ρ(π) Zariski-dense.

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Split R-forms and Hitchin representations Theorem (Hitchin 1992). Let G = PSL(n, R).

• Hom(π, G)/G has 3 or 6 components,

depending on whether n is odd or even, respectively (n > 2).

• ∃ a component H of Hom(π, G) which

is a cell of dimension 2(g − 1)(dim G) = 2(g − 1)(n2 − 1).

• H contains ι ◦ φ,

π

φ

֒ → SL(2, R)

ι

− → G where φ Fuchsian, and ι the irreducible n-dimensional representation of SL(2, R).

• H identifies with

H0(Σ, κ2) ⊕ H0(Σ, κ3) ⊕ · · · ⊕ H0(Σ, κn). A representation in Hitchin’s component H is called a Hitchin representation.

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Convex RP2-structures When G = SL(3, R), Hitchin representa- tions enjoy special geometric significance: Theorem (Choi-Goldman 1993). Hitchin representations π

ρ

− → SL(3, R) ← → marked convex RP2-structures on S, that is, diffeomorphisms S

≈

− → Ω/Γ where Ω ⊂ RP2 is a convex domain and Γ ⊂ Aut(Ω) is a discrete group acting properly on Ω. ∂Ω is a C1+α strictly convex curve, where 0 < α ≤ 1. α = 1 ⇐ ⇒ ∂Ω is a conic, and ρ equals π

φ

֒ → SO(2, 1) ֒ → SL(3, R) where φ is Fuchsian.

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Higgs bundles and affine spheres Higgs bundle theory = ⇒ H identifies with the vector space H0(Σ, κ2) ⊕ H0(Σ, κ3) ∼ = C8g−8 but this identification depends strongly on the conformal structure Σ. Theorem (Labourie 1997, Loftin 1999). The space of marked convex RP2-structures naturally identifies with the holomorphic vector bundle over TS whose fiber over Σ is H0(Σ, κ3). Follows Calabi, Loewner-Nirenberg, Cheng- Yau, Gigena, Sasaki, Li, Wang on hyper- bolic affine spheres. For every Hitchin representation, ∃ a unique conformal structure so that ˜ Σ

˜ h

− → G/K is a conformal map, that is the component

• f the Higgs field in H0(Σ, κ2) — the Hopf

differential of h — vanishes.

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Hyperconvex representations In 2002, Labourie found a dynamical/geometric characterization of Hitchin representations. Theorem (Labourie). A Hitchin represen- tation in SL(n, R) is a discrete quasi-isometric embedding π

ρ

֒ → SL(n, R) with reductive image. Also true for maximal representations. A curve S1 f − → RPn−1 is hyperconvex :⇐ ⇒ ∀x1, . . . , xn ∈ S1 distinct, f(x1) + · · · + f(xn) = Rn.

• Theorem. (Guichard-Labourie 2006)

ρ is Hitchin ⇐ ⇒ preserves hyperconvex curve. ∂Σ = ∅: Fock-Goncharov. When n = 4, Guichard and Wienhard as- sociate to a Hitchin representation an RP3- structure on the unit tangent bundle T1(S).

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Speculation Particularly interesting: Sp(2n, R), both:

• R-split (so ∃ a Hitchin component);
• G/K Hermitian-symmetric tube do-

main (Siegel upper-half space). Higgs bundle techniques = ⇒ topology of de- formation space. (B-G-G-Mundet i Riera.) Hitchin representations are maximal, although ∃ other maximal representations. Example: G = Sp(4, R) ∼ SO(3, 2)

• Hitchin representations deform Fuchsian

representations π

φ

֒ → SL(2, R) − → Aut

• Sym3(R2)
• ⊂ Sp(4, R);
• Other maximal representations deform

diagonally embedded Fuchsian π

φ

֒ → SL(2, R)

∆

− → SL(2, R)×SL(2, R) ⊂ Sp(4, R).

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Symmetric space G/K comprises positive compatible complex structures on 4-dimensional symplectic R-vector space. Minimal parabolic G-homogeneous spaces:

• Projective space P(R4) with PSp(4, R)-

invariant contact structure;

• Einstein compactification Ein2,1 of R2,1

with a Lorentz-conformal structure.

• Problem. Given maximal rep into Sp(4, R) ∼

SO(3, 2), find conformal structure Σ such that the harmonic map into G/K is con- formal. Interpret these representations as geometric structures on T1(S):

• Contact 1-forms and CR-structures

compatible with contact RP3-structures modeled on

• PSp(4, R), P(R4)
• ;
• Conformally flat Lorentz metrics

modeled on

• SO(3, 2), Ein2,1

Would identify H with the bundle over TS with fiber H0(Σ, κ4) analogous to Labourie- Loftin parametrization of SL(3, R)-Hitchin component by H0(Σ, κ3).