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. 1

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

3 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 π 0 ( 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 H 2 ; • G = PSL (2 , C ), orientation-preserving isometries of H 3 ; • G = SU ( n, 1), holomorphic isometries of H n C ; • G = PSL ( n + 1 , R ), collineations of R P n ; • G = Sp (4) ∼ SO (3 , 2).

4 Hyperbolic structures on surfaces The following constructions are equivalent: • Curvature − 1 Riemannian metric on S ; • Alas of H 2 -valued coordinate charts on S , locally isometric coordinate changes; • A pair ( ρ, dev ), representation ρ → PSL (2 , R ) , − π equivariant immersion dev ˜ → H 2 ; S − • ( ρ, dev ) with dev isometry; • ( ρ, dev ) with ρ Fuchsian

5 Fuchsian representations of surface groups ρ is Fuchsian : � • Embedding with discrete image; � • Free proper isometric π -action on H 2 ; � • ρ ( γ ) is hyperbolic ∀ γ � = 1. � hyperbolic projective actions on R P 1 . Fuchsian representations comprise connected components of Hom ( π, PSL (2 , R )) (Weil 1960). Uniformization theorem establishes equiv- alence between (equivalence classes of): � Fuchsian reps � � Marked conformal � ← → structures Σ ≈ S π − → PSL (2 , R ) which is the Teichm¨ uller space T S of S .

6 Characteristic classes Euler class of associated H 2 -bundle defines: → H 2 ( S ; Z ) ∼ Euler − − Hom ( π 1 ( S ) , G ) = Z satisfying | Euler ( ρ ) | ≤ | χ ( S ) | = 2 g − 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 ρ flat X -bundle with holonomy π − → G = ⇒ τ Hom ( π, G ) − → Z satisfying | τ ( ρ ) | ≤ (2 g − 2) rank R ( G ) . (Domic-Toledo 1987) The invariant is a bounded cohomology class in H 2 ( G ) ρ is maximal : ⇐ ⇒ | τ ( ρ ) | = (2 g − 2) rank R ( G ) .

7 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 4 g − 3 preimages R e := { ρ | Euler ( ρ ) = e } for e = 2 − 2 g, 3 − 2 g, . . . , 0 , . . . , 2 g − 2 . Maximal representations thus correspond to hyperbolic structures on S . What do representations in other compo- nent correspond to?

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9 Some examples • Take a regular octagon with angles π/ 4. Pair the sides by a 1 , b 1 , a 2 , b 2 ∈ PSL (2 , R ) according to the pattern below. Around the single vertex in the quotient is a cone of angle 8( π/ 4) = 2 π, a disc in the hyperbolic plane. These isometries satisfying the defining rela- tion for π 1 ( S ): a 1 b 1 a − 1 1 b − 1 1 a 2 b 2 a − 1 2 b − 1 = 1 2 and therefore defines a representation ρ π 1 ( S ) − → 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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� � �� 11 • Take a regular right-angled octagon. Again, side pairings a 1 , b 1 , a 2 , b 2 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 of cone angle 4 π = 8( π/ 2) . Since the product a 1 b 1 a − 1 1 b − 1 1 a 2 b 2 a − 1 2 b − 1 2 is rotation through 4 π (the identity), the holonomy representation ˆ ρ of the nonsingular hyperbolic surface S \ { p } extends: � � S \ { p } π 1 ρ ˆ � � � � � � � � � � � � � π 1 ( S ) PSL (2 , R ) � � � � ρ � � and Euler ( ρ ) = − 1. Although ρ ( π ) is discrete, ρ is not injective.

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13 • Take a hyperbolic surface M with ho- lonomy ψ π 1 ( M ) − → PSL (2 , R ) and a degree-one map f S − → M not homotopic to a homeomorphism. (For example, collapse ≥ 1 handles). The composition ψ f ∗ π = π 1 ( S ) − → π 1 ( M ) ֒ → 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.)

14 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 , rank ( W ) < deg ( V ) deg ( W ) rank ( V ) . Theorem (Hitchin-Donaldson-Corlette-Simpson) . There is a natural bijection between equiv- alences classes: � � � Stable Higgs pairs � Irreducible rep’ns ← → ρ ( V, Φ) over Σ π 1 (Σ) − → 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 of compact Riemannian manifolds. (Eels-Sampson)

15 Harmonic metrics Going from ρ to ( V, Φ) involves finding a harmonic metric, Hermitian metric h on the flat C n -bundle E ρ with holonomy ρ , whose corresponding ρ -equivariant map � h � − → GL ( n, C ) / U ( n ) M is harmonic map into the symmetric space. Harmonic metric h determines Higgs pair: → ∂h ∈ Ω 1 � � • Higgs field Φ ← End ( V ) . • Holomorphic structure ¯ ∂ V on V arises from conformal structure Σ and the unique Hermitian connection ∇ h preserving h . The self-duality equations: ( d A ) ′′ (Φ) = 0 F ( A ) + [Φ , Φ ∗ ] = 0 A is h -unitary connection, F ( A ) curvature, Φ ∗ adjoint of Φ with respect to h .

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

17 Then ( V, Φ) stable Higgs pair with Higgs field corresponding to � � 0 s D Φ = . Q 0 where: • s D meromorphic section of D , giving Higgs field in κ ⊗ Hom ( L 2 , L 1 ) ∼ = D ⊂ Ω 1 � �� Σ , End ( V ) ; • Q ∈ H 0 (Σ , κ 2 ) holomorphic quadratic differential with div ( Q ) ≥ D. determining Higgs field in κ ⊗ Hom ( L 1 , L 2 ) ∼ = κ 2 ⊂ Ω 1 � �� 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.

18 Symmetric products The Euler class e relates to the degree d by: e = 2 − 2 g + d Theorem (Hitchin 1987) . R e identifies with a holomorphic vector bundle over the sym- metric power Sym d (Σ) . The fiber over D ∈ Sym d (Σ) is the 3 g − 3 − d -dimensional vector space { Q ∈ H 0 (Σ , κ 2 ) | div ( Q ) ≥ D } . Q ← → Hopf differential of h . When e = 2 − 2 g , then d = 0 and the space ( T S ) of Fuchsian representations identifies with the vector space H 0 (Σ , κ 2 ). 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 2 π .

19 Uniformization with singularities Branched conformal structures on S with cone angles θ 1 , . . . , θ k ∈ 2 π Z + form bundle S d of symmetric powers Sym d (Σ) over T S where k � d = 1 ( θ i − 2 π ) . 2 π i =1 Theorem (Hitchin, McOwen, Troyanov) . Given branched conformal structure (Σ , D ) with k � 2 − 2 g + ( θ i − 2 π ) > 0 , i =1 ∃ ! branched hyperbolic structure confor- mal to (Σ , D ) . Resulting uniformization map S d U − → H 2 − 2 g + d ⊂ Hom ( π, G ) /G is homotopy equivalence. (Hitchin) But it’s not surjective!