fenchel-nielsen coordinates - trouser decomposition of riemann
TRANSCRIPT
Classification of SurfacesRiemann Surfaces
Simple Closed Curves on SurfacesFenchel-Nielsen coordinates
Fenchel-Nielsen CoordinatesTrouser Decomposition of Riemann surfaces
Hyungryul Baik
Department of MathematicsCornell University
Olivetti Club, Nov. 2010
Harry Baik Fenchel-Nielsen Coordinates
Classification of SurfacesRiemann Surfaces
Simple Closed Curves on SurfacesFenchel-Nielsen coordinates
Beginning Question
What is a surface?
Harry Baik Fenchel-Nielsen Coordinates
Classification of SurfacesRiemann Surfaces
Simple Closed Curves on SurfacesFenchel-Nielsen coordinates
Examples of surfaces
Harry Baik Fenchel-Nielsen Coordinates
Classification of SurfacesRiemann Surfaces
Simple Closed Curves on SurfacesFenchel-Nielsen coordinates
Classification of compact surfaces
How many different surfaces are there?How do we classify them?
Harry Baik Fenchel-Nielsen Coordinates
Classification of SurfacesRiemann Surfaces
Simple Closed Curves on SurfacesFenchel-Nielsen coordinates
Classification of compact surfaces
Fact: Any compact connected 2-manifold can be obtained froma polygon in the plane by gluing corresponding sides of theboundary together.For example, the g-holed torus T 2#T 2# . . .T 2 can berepresented by 4g-gon by identifying edges using symbola1b1a−1
1 b−11 . . . agbga−1
g b−1g .
Harry Baik Fenchel-Nielsen Coordinates
Classification of SurfacesRiemann Surfaces
Simple Closed Curves on SurfacesFenchel-Nielsen coordinates
Classification of compact surfaces
Harry Baik Fenchel-Nielsen Coordinates
Classification of SurfacesRiemann Surfaces
Simple Closed Curves on SurfacesFenchel-Nielsen coordinates
Classification of compact surfaces
Harry Baik Fenchel-Nielsen Coordinates
Classification of SurfacesRiemann Surfaces
Simple Closed Curves on SurfacesFenchel-Nielsen coordinates
They are all we need
Conway¡¯s ZIP Proof, George K. Francis Jeffrey R. WeeksHarry Baik Fenchel-Nielsen Coordinates
Classification of SurfacesRiemann Surfaces
Simple Closed Curves on SurfacesFenchel-Nielsen coordinates
Actually, more than enough
Lemma
A crosshandle is homeomorphic to two crosscaps.
Harry Baik Fenchel-Nielsen Coordinates
Classification of SurfacesRiemann Surfaces
Simple Closed Curves on SurfacesFenchel-Nielsen coordinates
Classification of compact surfaces
Harry Baik Fenchel-Nielsen Coordinates
Classification of SurfacesRiemann Surfaces
Simple Closed Curves on SurfacesFenchel-Nielsen coordinates
Classification of surfaces
Dyck’s Theorem
Handles and crosshandles are equivalent in the presenceof a crosscap.
Harry Baik Fenchel-Nielsen Coordinates
Classification of SurfacesRiemann Surfaces
Simple Closed Curves on SurfacesFenchel-Nielsen coordinates
Classification of surfaces
Harry Baik Fenchel-Nielsen Coordinates
Classification of SurfacesRiemann Surfaces
Simple Closed Curves on SurfacesFenchel-Nielsen coordinates
Classification of surfaces
Let a be the connected sum with a torus (or putting a handle)and let b be the connected sum with RP2 (or putting across-cap). Then this is a sort of semigroup generated by a andb with a relation ba = b3.
Harry Baik Fenchel-Nielsen Coordinates
Classification of SurfacesRiemann Surfaces
Simple Closed Curves on SurfacesFenchel-Nielsen coordinates
Riemann surface
Definition #1 Riemann surface
A Riemann surface is a one-complex-dimensionalconnected complex manifold; that is, a
two-real-dimensional connected manifold M with amaximal set of charts {Uα, φα}α∈A on M such that
φα : Uα → C is a homeomorphism onto an open subset ofthe complex plane C and the transition functions are
holomorphic maps.
Harry Baik Fenchel-Nielsen Coordinates
Classification of SurfacesRiemann Surfaces
Simple Closed Curves on SurfacesFenchel-Nielsen coordinates
Uniformization Theorem
Theorem #1 Uniformization theorem for Riemann surfaces
A simply connected Riemann surface is isomorphic toeither the Riemann sphere P1, then complex plane C, or
the open unit disc D ⊂ C.
Harry Baik Fenchel-Nielsen Coordinates
Classification of SurfacesRiemann Surfaces
Simple Closed Curves on SurfacesFenchel-Nielsen coordinates
Möbius transformation
Definition #2 Möbius transformation
A Möbius transformation is a mapping P1 → P1 of the formz 7−→ ax+b
cz+d with ad − bc 6= 0.
Theorem #2 Automorphisms of simply connected Riemannsurfaces
Aut(P1) is the group of all Möbius transformations.Aut(C) is the group of Möbius transformations of the formz 7−→ az + b with a 6= 0.Aut(D) is the group of Möbius transformations of the formz 7−→ az+b
bz+awith |a|2 > |b|2.
Harry Baik Fenchel-Nielsen Coordinates
Classification of SurfacesRiemann Surfaces
Simple Closed Curves on SurfacesFenchel-Nielsen coordinates
Möbius transformation
Definition #2 Möbius transformation
A Möbius transformation is a mapping P1 → P1 of the formz 7−→ ax+b
cz+d with ad − bc 6= 0.
Theorem #2 Automorphisms of simply connected Riemannsurfaces
Aut(P1) is the group of all Möbius transformations.Aut(C) is the group of Möbius transformations of the formz 7−→ az + b with a 6= 0.Aut(D) is the group of Möbius transformations of the formz 7−→ az+b
bz+awith |a|2 > |b|2.
Harry Baik Fenchel-Nielsen Coordinates
Classification of SurfacesRiemann Surfaces
Simple Closed Curves on SurfacesFenchel-Nielsen coordinates
Universal covering space of Riemann surfaces
Theorem #3 Universal covering space of Riemann surfaces
The Riemann sphere P1 is the universal cover only of itself.The plane C is the universal cover of itself, of thepunctured plane C− {a}, and of all compact Riemannsurfaces homeomorphic to a torus.All other Riemann surfaces have universal covering spaceanalytically isomorphic to D.
Definition #3 Hyperbolic Riemann surface
A Riemann surface is hyperbolic if its universal coveringspace is isomorphic to D.
Harry Baik Fenchel-Nielsen Coordinates
Classification of SurfacesRiemann Surfaces
Simple Closed Curves on SurfacesFenchel-Nielsen coordinates
Universal covering space of Riemann surfaces
Theorem #3 Universal covering space of Riemann surfaces
The Riemann sphere P1 is the universal cover only of itself.The plane C is the universal cover of itself, of thepunctured plane C− {a}, and of all compact Riemannsurfaces homeomorphic to a torus.All other Riemann surfaces have universal covering spaceanalytically isomorphic to D.
Definition #3 Hyperbolic Riemann surface
A Riemann surface is hyperbolic if its universal coveringspace is isomorphic to D.
Harry Baik Fenchel-Nielsen Coordinates
Classification of SurfacesRiemann Surfaces
Simple Closed Curves on SurfacesFenchel-Nielsen coordinates
What do we know about hyperbolic Riemannsurfaces?
A Fuchsian group is a discrete subgroup of Aut(H2), whichis isomorphic PSL2(R).A hyperbolic Riemann surface X could be identified withH2/Γ for some Fuchsian group Γ which is isomorphic toπ1(X ).A Fuchsian group acts on H2 properly discontinuously.(Conversely, any subgroup of Isom+(H2) which acts on H2
properly discontinuously is discrete.)
Harry Baik Fenchel-Nielsen Coordinates
Classification of SurfacesRiemann Surfaces
Simple Closed Curves on SurfacesFenchel-Nielsen coordinates
What do we know about hyperbolic Riemannsurfaces?
Theorem #4 Universal covering space of Riemann surfaces
If a Fuchsian group Γ is torsion-free, then X := H2/Γ hasa unique structure of a Riemann surface for which the
projection is a local isomorphism.
So, the torsion-free Fuchsian groups and hyperbolic Riemannsurfaces are essentially the same subject.
Harry Baik Fenchel-Nielsen Coordinates
Classification of SurfacesRiemann Surfaces
Simple Closed Curves on SurfacesFenchel-Nielsen coordinates
Essential curves
Let X be a surface and γ : [a,b]→ X be a curve on X .γ is simple if it is injective.γ is closed if γ(a) = γ(b).γ is essential if γ is not homotopic to a point or a boundarycomponent, if there is any.
Harry Baik Fenchel-Nielsen Coordinates
Classification of SurfacesRiemann Surfaces
Simple Closed Curves on SurfacesFenchel-Nielsen coordinates
Geodesic representative
Proposition Geodesic representative
For an essential simple closed curve c, there exists aunique geodesic γ in the free homotopy class of c.
Why? Let’s lift c to the universal cover and look for the decktransformation which maps one end point to another.Is it elliptic? Parabolic? Hyperbolic? Wait, what are they?
Harry Baik Fenchel-Nielsen Coordinates
Classification of SurfacesRiemann Surfaces
Simple Closed Curves on SurfacesFenchel-Nielsen coordinates
Isometries of Hyperbolic plane
Let f be an isometry of hyperbolic plane.
f is elliptic if it fixes one point in the hyperbolic plane.f is parabolic if it fixes one point on the boundary ofhyperbolic plane.f is hyperbolic if it fixes two points on the boundary ofhyperbolic plane.
Harry Baik Fenchel-Nielsen Coordinates
Classification of SurfacesRiemann Surfaces
Simple Closed Curves on SurfacesFenchel-Nielsen coordinates
Maximal Essential curves
Cut your Riemann surface along a family of disjoint closedgeodesics. Until when?
Definition #5 filling
A family F of essential simple closed curves on thesurface S is said to fill S if any other essential simple
closed curve should intersects an element of F .
When this family fills the surface, call it a loop system. And bythe previous result, we may assume that each element in theloop systems is a closed geodesic.
Harry Baik Fenchel-Nielsen Coordinates
Classification of SurfacesRiemann Surfaces
Simple Closed Curves on SurfacesFenchel-Nielsen coordinates
Decomposition
What the complement of a loop system looks like?
Harry Baik Fenchel-Nielsen Coordinates
Classification of SurfacesRiemann Surfaces
Simple Closed Curves on SurfacesFenchel-Nielsen coordinates
Decomposition
Harry Baik Fenchel-Nielsen Coordinates
Classification of SurfacesRiemann Surfaces
Simple Closed Curves on SurfacesFenchel-Nielsen coordinates
Main Theorem
Theorem #5 Pants-decomposition
A hyperbolic Riemann surface R of genus g withoutboundary always contains a loop system of 3g − 3 disjoint
simple closed geodesics. Regardless of which loopsystem we choose, cutting R along the geodesics in thesystem always decomposes R into 2g − 2 pairs of pants.
Proof? First, let’s assume that each component of R aftercutting is a pair of pants, and count the number of pairs ofpants in the maximal decomposition.
Harry Baik Fenchel-Nielsen Coordinates
Classification of SurfacesRiemann Surfaces
Simple Closed Curves on SurfacesFenchel-Nielsen coordinates
Maximal Essential curves
N = the number of disjoint simple closed geodesics in anysuch system on RM = the number of pairs of pants in a decomposition of RL = an element of our loop systemn1 = the number of connected components of R − Lg1 = the sum of the genera of the connected componentsof R − L
Then g1 − n1 = (g − 1)− 1 and R − L has two boundarycomponents.
Harry Baik Fenchel-Nielsen Coordinates
Classification of SurfacesRiemann Surfaces
Simple Closed Curves on SurfacesFenchel-Nielsen coordinates
We proceed inductively, cutting R successively along loops inour set of disjoint simple closed geodesics. Whenever we cutalong a new loop,
the resulting set has two more boundary components,the sum of the genera less the number of connectedcomponents decreases by one.
Harry Baik Fenchel-Nielsen Coordinates
Classification of SurfacesRiemann Surfaces
Simple Closed Curves on SurfacesFenchel-Nielsen coordinates
Thus after we cut along N closed geodesics, we get
3M = 2N and gN − nN = (g − 1)− N.
But gN = 0 and nN = M so that M = N − g + 1.Then 2N = 3M = 3N − 3g + 3 so that
N = 3g − 3 and M = 2g − 2.
Harry Baik Fenchel-Nielsen Coordinates
Classification of SurfacesRiemann Surfaces
Simple Closed Curves on SurfacesFenchel-Nielsen coordinates
A pair of pants
Definition #6 A pair of pants (or a trouser)
A pair of pants is a complete hyperbolic surface withgeodesic boundary, whose interior is homeomorphic to
the complement of three points in the 2-sphere.
Harry Baik Fenchel-Nielsen Coordinates
Classification of SurfacesRiemann Surfaces
Simple Closed Curves on SurfacesFenchel-Nielsen coordinates
A pair of pants!
Harry Baik Fenchel-Nielsen Coordinates
Classification of SurfacesRiemann Surfaces
Simple Closed Curves on SurfacesFenchel-Nielsen coordinates
Why do we care about trousers?
Proposition
Let X be a compact connected hyperbolic surface withgeodesic boundary. If all simple closed geodesics of X
are boundary components, then X is homeomorphic to apair of pants.
This tells us why pairs of pants are natural building blocks forRiemann surfaces: they are the only compact hyperbolicsurfaces with geodesic boundary that cannot be furthersimplified by cutting along simple geodesics.
Harry Baik Fenchel-Nielsen Coordinates
Classification of SurfacesRiemann Surfaces
Simple Closed Curves on SurfacesFenchel-Nielsen coordinates
Sketch
Consider various number of boundary components of X . 0, 1,or 2..In case, where X has at least two boundary components A,B,let C be be a simple arc joining A and B.Intersection number? why geodesics are in minimal position?
Harry Baik Fenchel-Nielsen Coordinates
Classification of SurfacesRiemann Surfaces
Simple Closed Curves on SurfacesFenchel-Nielsen coordinates
Non-compact version
Proposition
Let X be a noncompact connected hyperbolic surface withcompact geodesic boundary, perhaps empty; assume that
every simple closed curve in X is either homotopic to apoint, or bounds a punctured disc, or is homotopic to a
boundary component. Then one of the following sixpossibilities holds:
X is a trouser with one, two, or three cuspsX is a half-annulus {z ∈ C|1 ≥ |z| < R} for some1 < R <∞X is isomorphic to the punctured disc D∗
X is isomorphic to D.
Proof is similar with the previous one.Harry Baik Fenchel-Nielsen Coordinates
Classification of SurfacesRiemann Surfaces
Simple Closed Curves on SurfacesFenchel-Nielsen coordinates
hyperbolic hexagons
First note that there is a unique common perpendicular to twogeodesics which do not intersect (even on the boundary) inhyperbolic plane.Let’s lift two boundary components of a trouser. What can weknow from this?
Harry Baik Fenchel-Nielsen Coordinates
Classification of SurfacesRiemann Surfaces
Simple Closed Curves on SurfacesFenchel-Nielsen coordinates
A pair of pants with three seams
Harry Baik Fenchel-Nielsen Coordinates
Classification of SurfacesRiemann Surfaces
Simple Closed Curves on SurfacesFenchel-Nielsen coordinates
Right-angled hexagon
Proposition
For any given positive numbers l1, l2, l3, there exists aunique hyperbolic right-angled hexagon with alternating
edge lengths (l1, l2, l3).
Harry Baik Fenchel-Nielsen Coordinates
Classification of SurfacesRiemann Surfaces
Simple Closed Curves on SurfacesFenchel-Nielsen coordinates
A pair of pants is decomposed into two right-angledhexagons
Harry Baik Fenchel-Nielsen Coordinates
Classification of SurfacesRiemann Surfaces
Simple Closed Curves on SurfacesFenchel-Nielsen coordinates
A pair of pants is decomposed into two right-angledhexagons
Harry Baik Fenchel-Nielsen Coordinates
Classification of SurfacesRiemann Surfaces
Simple Closed Curves on SurfacesFenchel-Nielsen coordinates
What’s left?
What did we get? Let X be a hyperbolic Riemann surface withgenus g having no boundary.
We have a loop system with 3g − 3 simple closedgeodesics.Each connected component of the complement of the loopsystem is a trouser.The hyperbolic structure of a trouser is completelydetermined by the lengths of boundary components (orcuffs!).
How about the hyperbolic structure of the entire Riemannsurface?
Harry Baik Fenchel-Nielsen Coordinates
Classification of SurfacesRiemann Surfaces
Simple Closed Curves on SurfacesFenchel-Nielsen coordinates
Dehn Twist
f : S × I → S × I, (s, t) 7→ (se2πit , t).
Harry Baik Fenchel-Nielsen Coordinates
Classification of SurfacesRiemann Surfaces
Simple Closed Curves on SurfacesFenchel-Nielsen coordinates
Dehn Twist
Harry Baik Fenchel-Nielsen Coordinates
Classification of SurfacesRiemann Surfaces
Simple Closed Curves on SurfacesFenchel-Nielsen coordinates
Teichmüller Space
Let S be a closed, oriented surface of genus g ≥ 2. A markedhyperbolic surface is a pair (φ,X ) consisting of an orientedcompact hyperbolic surface X ∼= H/Γ and anorientation-preserving homeomorphism φ : S → X .Two marked surfaces (φi ,Xi), i = 1,2 are equivalent if thereexists an isometry α : X1 → X2 such that φ−1
2 ◦ α ◦ φ1 = ψ isisotopic to the identity.
Definition #6 Teichmüller Space
The space of such equivalence classes is theTeichmüller space Tg = Teich(S).
Harry Baik Fenchel-Nielsen Coordinates
Classification of SurfacesRiemann Surfaces
Simple Closed Curves on SurfacesFenchel-Nielsen coordinates
Fenchel-Nielsen Coordinates
Intuitively, what we have shown is that any point Teichmüllerspace of a hyperbolic Riemann surface of genus g may bespecified by 3g − 3 nonnegative real numbers (lengths ofsimple closed geodesics) and 3g − 3 real numbers (degree oftwisting between glued pairs of pants). Thus Tg is equivalent(as sets) to
R3g−3+ × R3g−3.
Harry Baik Fenchel-Nielsen Coordinates
Classification of SurfacesRiemann Surfaces
Simple Closed Curves on SurfacesFenchel-Nielsen coordinates
Pants-decomposition revisited
But the pants-decomposition of a surface is not unique!
Harry Baik Fenchel-Nielsen Coordinates
Classification of SurfacesRiemann Surfaces
Simple Closed Curves on SurfacesFenchel-Nielsen coordinates
Coordinate Change
Harry Baik Fenchel-Nielsen Coordinates
Classification of SurfacesRiemann Surfaces
Simple Closed Curves on SurfacesFenchel-Nielsen coordinates
Coordinate Change
Harry Baik Fenchel-Nielsen Coordinates
Classification of SurfacesRiemann Surfaces
Simple Closed Curves on SurfacesFenchel-Nielsen coordinates
Moduli spaces and Mapping Class groups
We let Mod(S) denote the group of orientation-preservinghomeomorphisms ψ : S → S, modulo those isotopic(equivalently, homotopic) to the identity. It acts on Teich(S) byψ · (φ,X ) = (φ ◦ ψ−1,X ).The quotient space is the moduli space
Mg =M(S) = Teich(S)/Mod(S).
Let S be the space of complex structures on S.M(S) = S/Homeo+(S)
T (S) = S/Homeo0(S)
Harry Baik Fenchel-Nielsen Coordinates
Classification of SurfacesRiemann Surfaces
Simple Closed Curves on SurfacesFenchel-Nielsen coordinates
Hmm
(If time permits) Collar Theorem
Harry Baik Fenchel-Nielsen Coordinates
Classification of SurfacesRiemann Surfaces
Simple Closed Curves on SurfacesFenchel-Nielsen coordinates
Moduli spaces are symplectic
Let li and ti be denote the length coordinates and twistingcoordinates, respectively.They are not well-defined on the Moduli space, but theirderivatives are:define the 2-form on Teichmüller space ω =
∑i dli ∧ dti .
Wolpert showed that this 2-form does not depend on the choiceof coordinates, so it descends to a 2-form on Moduli space.
Harry Baik Fenchel-Nielsen Coordinates
Classification of SurfacesRiemann Surfaces
Simple Closed Curves on SurfacesFenchel-Nielsen coordinates
Few words on Complex analytic theory
Conformal vs. quasi-conformal maps
Beltrami differental µ(z)dzdz
M(R) - space of Beltrami differentials with ||µ||∞ < 1.{conformal classes of quasi-conformal deformations of agiven Riemann surface} ↔ { elements in M(R)}.Two quasiconformal maps f0 : R → R0, f1 : R → R1 areTeichm"uller equivalent if there exists a conformal mapc : R0 → R1 and f−1
1 ◦ c ◦ f0 is homotopic to the identity onR such that boundary of R is fixed point-wise underhomotopy.The space of equivalence classes in M(R) with aboverelation is T (R).
Harry Baik Fenchel-Nielsen Coordinates
Classification of SurfacesRiemann Surfaces
Simple Closed Curves on SurfacesFenchel-Nielsen coordinates
References
Teichmüller theory and application to Geometry, Topology,and Dynamics, John. H. HubbardThree dimensional Geometry and Topology, William P.ThurstonOn groups generated by two positive multi-twists:Teichüller curves and Lehmer’s number, Christopher JLeiningerThe Fenchel-Nielsen Coordinates of Teichmüller Space,Kathy Paur
Harry Baik Fenchel-Nielsen Coordinates