Complex Projective Structures and the Bers Embedding August 3, 2003 - PDF document

Complex Projective Structures and the Bers Embedding August 3, 2003 David Dumas (ddumas@math.harvard.edu) http://www.math.harvard.edu/˜ddumas/

Plan • The Bers Embedding • Beyond the Bers Embedding • Grafting and Fuchsian Centers • Estimates on the Distribution 2

Bers embedding: β X : Teich( X ) ֒ → Q ( X ) Holomorphic embedding, image is a bounded do- main. (Good!) Depends on the choice of a basepoint – a complex structure X . Definition: β X ( Y ) = S ( f X,Y : ∆ → Ω + X,Y ), where: f X,Y : ∆ → Ω + X,Y is a Riemann map Ω + X,Y is the domain of discontinuity of qf ( X, Y ) with quotient RS X qf ( X, Y ) is the quasifuchsian group simultane- ously uniformizing X and Y � ′ � 2 � f ′′ � f ′′ − 1 S ( f ) = , the Schwarzian derivative f ′ f ′ 2 3

The Bers Embedding for the Hexagonal Torus. 4

Suppose β X ( Y ) = φ ∈ Q ( X ). The quad diff φ records the failure of the univalent function f : ∆ → Ω + X,Y to be M¨ obius. The quasifuchsian group qf ( X, Y ) is the holon- omy group of φ (i.e. of the ODE u ′′ + 1 2 φu = 0). In fact, any φ ∈ Q ( X ) determines a holonomy rep- resentation ρ : π 1 ( X ) → PSL 2 ( C ) and an equiv- ariant holomorphic map f : ∆ → ˆ C satisfying S ( f ) = φ . For large enough φ , f is not univalent. (It is locally univalent iff φ is L ∞ .) The holonomy group ρ ( π 1 ( X )) may not be dis- crete. (It could be anything.) But there are quad diffs φ (even opens sets of them) with discrete holonomy outside β X (Teich( X )). The Bers embedding into Q ( X ) is just one island in a vast archipelago! 5

Discrete Holonomy for the Hexagonal Torus. 6

Main Question: What does the set of φ ∈ Q ( X ) with discrete (or QF) holonomy look like? I.e. where are the islands of discrete holonomy, and what do they look like? Convenient to study this using CP 1 geometry: The pair ( X, φ ) determines: • f : ∆ → ˆ C , locally univalent, equivariant, S ( f ) = φ ; developing map • ρ : π 1 ( X ) → PSL 2 ( C ) holonomy These define a complex projective structure on X , i.e. an atlas of charts with M¨ obius transition functions. Q = { ( X, φ ) | X ∈ Teich , φ ∈ Q ( X ) } is the space of all projective surfaces. Hence, question becomes: Which projective struc- tures on X have discrete (QF) holonomy? 7

Why do we care? • Natural extension of study of the Bers em- bedding. • Topology of AH ( S ) – CP 1 structure give a geometric interpretation that is insensitive to discreteness of a representation. • Any (irreducible) representation ρ : π 1 ( S ) − → PSL 2 ( C ) arises from a CP 1 structure. (Gallo, Kapovich, Marden; Annals 2000) • Correspond to locally convex pleated surfaces in hyperbolic manifolds. 8

1. How many islands are there? Infinitely many islands (with disjoint interiors) appear in each Q ( X ). ( W. Goldman + H. Tanigawa) Idea: The Bers embedding has a natural cen- ter point – φ = 0 corresponding to the Fuchsian group qf ( X, X ). W. Goldman produces other examples of projec- tive structures with Fuchsian holonomy that are “exotic”, i.e. the developing map is not injective and thus they are outside the Bers embedding. The key is grafting , a cut-and-paste operation on hyperbolic surfaces. Start with Y ∈ Teich( X ) and a family of disjoint simple closed curves γ i . Cut Y along the hyper- bolic geodesics corresponding each γ i , and insert a tube of length h i . 9

This is grafting of Y along the weighted multic- urve α = � i h i γ i . The resulting surface is denoted gr α Y . Grafting yields more than just a surface; the result has a natural projective structure, Gr α Y , in which α is analogous to the bending of the convex core boundary in a quasifuchsian manifold. In fact, bending is a special case. 10

Goldman observes that grafting with weights in 2 π N gives Fuchsian holonomy, essentially because the developing map f wraps completely around ˆ C . (In fact, this is the only way to obtain Fuchsian holonomy.) Let M L Z denote the set multicurves with 2 π - integral weights. These are examples of measured laminations . By a result of H. Tanigawa, for each α ∈ M L Z there is a unique starting surface Y α such that the grafted surface is isomorphic to X . The projective structure on this grafted surface has Fuchsian holonomy qf ( Y α , Y α ); let φ ( α ) de- note its Schwarzian. 11

We call φ ( α ) the Fuchsian center with wrap- ping invariant α , since α encodes the way the developing map wraps around ˆ C . For α � = 0, φ ( α ) lies outside of the Bers embed- ding and provides a kind of center point for an island of QF holonomy that surrounds it. For topological reasons, different Fuchsian cen- ters must lie on different islands. This estab- lishes the answer to our question (“How many islands?”): different wrapping invariants ⇓ different islands ⇓ infinitely many islands 12

The association between projective surfaces and grafting is not limited to Fuchsian holonomy. Thurston has shown that every projective struc- ture arises from grafting in a unique way: ≃ Gr : M L × Teich � Q Here one must allow grafting along all measured laminations (which continuously interpolate be- tween simple closed curves). The definition of this kind of grafting is as a limit of the simple closed curve case. Scannell and Wolf: For fixed X and each λ ∈ M L , there is a unique Y λ ∈ Teich( X ) such that gr λ Y λ = X . The grafted projective structure gives φ ( λ ), generalizing φ ( α ). 13

2. Where are the islands located? We approach this problem by first trying to de- scribe the location of the Fuchsian centers, as they are in some ways simpler than general pro- jective structures. (Similarly, one usually studies Fuchsian groups before quasifuchsian groups.) For each Fuchsian center we have the following data: • the wrapping invariant α = � i 2 πn i γ i , a weighted multicurve ( M L Z ), → ˆ • the developing map f α : ∆ − C , • the Schwarzian φ ( α ) = S ( f α ), a quadratic dif- ferential on X , • a surface Y α from which X can be obtained by grafting: X = gr α Y α , • and the Fuchsian holonomy group qf ( Y α , Y α ) that uniformizes Y α . 14

Focus on two aspects of the question (“Where are the islands?”): 2a. Where is the Fuchsian center ( X, φ ( α )) ? 2b. What is Y α ? ( Y λ ?) ⇒ What is the holonomy of φ ( α ) ? ( φ ( λ )? ) ⇐ We can give partial answers to both questions and describe a more detailed conjectural picture. 15

First of all, what is a reasonable guess for the distribution of φ ( α )? If X is a punctured torus , a simple closed curve is uniquely determined by its slope, a primitive element of P H 1 ( X, Z ) ≃ ˆ Q / {± 1 } . A weighted simple closed curve is therefore spec- ified by a pair of integers ( m, n ) (corresponding to slope m/n with weight 2 π gcd( m, n )). In this case Q ( X ) ≃ C . 16

One might guess that { φ m,n } looks like Z [ i ]: Problem: ( m, n ) and ( − m, − n ) are supposed to represent the same curve.

Correct this by squaring the square lattice! (Apply z �→ z 2 to Z [ i ].) This approximate picture is consistent with what is known.

Discrete Holonomy for the Square Torus. 19

There is another natural holomorphic differential associated to α (or to any measured lamination) – the Strebel differential ψ ( α ). Defining property: When the horizontal foliation of ψ ( α ) is pulled taut on X , the result is the mea- sured geodesic lamination α . Thm A: Fix X and a weighted multicurve α . For each n ∈ N , φ ( nα ) = ψ ( nα ) + O α (1) , i.e. Fuchsian centers are near Strebel centers along each ray. In particular, � φ ( nα ) � 1 = E ( nα, X ) + O α (1) = O ( n 2 ) ; ⇒ quadratic growth. E ( α, X ) is the extremal length of α on X . Note: A quadratic lower bound for single curves was given by Anderson (Ph.D. thesis, Berkeley, 1998). 20

Conjecture: φ ( α ) ≈ ψ ( α ) The key here is to find a comparison between φ ( α ) and ψ ( α ) that is uniform across different sets of curves. More fundamentally, one might expect a connec- tion between the grafting lamination and Strebel differentials: Conjecture: φ ( λ ) ≈ ψ ( λ ) This would probably follow by continuity from any technique that addresses the previous conjecture. Very optimistic: a ≈ b − → a − b = O (1) Quite likely: a ≈ b a − b = o ( | a | , | b | ) − → 21

2b. Describe Y λ / the holonomy of φ ( λ ) . This amounts to inverting the operation of graft- ing. While this is difficult for any particular λ and X , limiting information can be extracted: Thm B: Fix X , and suppose ( X, φ ( λ )) = Gr λ Y λ . Then 1. ℓ ( λ, Y λ ) = E ( λ, X ) + O (1) ⇒ For multicurves, lengths on Y λ grow linearly. 2. If λ n → ∞ , while [ λ n ] → λ ∈ P M L , then Y λ → [ λ ′ ] ∈ ∂ Th Teich( X ) where λ ′ ∈ M L is associated to the vertical foliation of ψ ( λ ) ∈ Q ( X ) . One might say that ( λ, λ ′ ) are “orthogonal”, in that they define the horizontal and vertical folia- tions of a single holomorphic quadratic differential ( ψ ( λ ) = − ψ ( λ ′ )) But ,this notion depends sensitively on the base surface X . 22