What is the Euler characteristic of moduli space? Compactified moduli space? The Gauss-Bonnet theorem for cone manifolds and volumes of moduli spaces

Size: px

Start display at page:

Download "What is the Euler characteristic of moduli space? Compactified moduli space? The Gauss-Bonnet theorem for cone manifolds and volumes of moduli spaces"

Anissa Dixon
5 years ago
Views:

1 The Gauss-Bonnet theorem for cone manifolds and volumes of moduli spaces Illustration F Curtis McMullen Kappes--Möller (2012) Thurston (199) Allendoerfer--Weil (1943) Schwarz, Picard, Deligne--Mostow, Cohen--Wolfart, Parker, Sauter,... What is the Euler characteristic of moduli space? M 0,n = {(b 1,...,b n ) 2 b C n Fibration ( 0,n 1 )= (n 3) 0,n 1! M 0,n : b i 6= b j }/ Aut b C M 0,n 1 (M 0,n )=( 1) n+1 (n 3)! Theorem M 0,n 6= CH n 3 / for n>4. manifold Γ Γ Figure 6. A holomorphic map from an ideal pentagon to an ideal star. Avg Holomorphic of F 2 in the Galois hyperbolic conjugation metric? Answer: 1/3 Curves on surfaces. Let f : V MF 2 be a Teichmüller /2 curve generated by (X, ω 2 ). Assume that SL(X, ω) hasrealquadratictracefieldk R. Let k k denote the Galois involution1-1 of K/Q, sendinga + b d to a b d, 2 2 and let g g denote the corresponding involution SL 2 (K). orbifold cone manifold We have seen that Jac(X t ), t = f(t), admits real multiplication by K for all t. Let -1+1/2+1/ Σ = H/Γ, = -3/10 Γ SL 2 (K), be the -1+1/2+2/ Hilbert modular = -1/10 surface parameterizing all Abelian varieties with the given action of K, asin 6. Then we obtain a commutative diagram H V f M 2 σ n (M 0,n ) Compactified moduli space? H 2 Σ A (10.1) as in Corollary.3. If we make a change of basepoint for V,byreplacing(X, ω)witha (X, ω), (M 0,n ) A SL 2 (R), then f(v ) M 2 does not change and GL(X, ω) variesonlyby conjugation by A. Themainobservationofthissectionis: Theorem 10.1 After a suitable change of basepoint, we have GL(X, ω) GL 2 (K) and Example: σ(t) =(t, n= F (t)), wheref : H H satisfies F (g t) =g F (t) (P 1 P 1 P P 0 )= = 2 34 (P 1 P 1 +3P 1 3P 0 )=4+6 3=

Generating functions Corollary 2 (Getzler) The generating functions f(x) =x n=2 χ(m 0,n+1 ) xn n! are formal inverses of one another. and g(x) =x + Universal; via stable trees (M, L Ens. math.

2 Generating functions Corollary 2 (Getzler) The generating functions f(x) =x n=2 χ(m 0,n+1 ) xn n! are formal inverses of one another. and g(x) =x + Universal; via stable trees (M, L Ens. math.) n=2 χ(m 0,n+1 ) xn n! Moduli spaces of polyhedra... Fix µ 1,...,µ n, 0 <µ i < 1, P µ i = 2. Any (b i ) determines a meromorphic 1-form on b C:! = dx Q n 1 (x b i) µ i (!) = X µ i b i divisor of degree 2 ( b C,! ) = convex polyhedron in R 3 Cone angles 2 (1 µ i ). Example: n=12, μi = 1/6 n=4 n=... are complex hyperbolic after all Example: μ = (,,,,)/1 M 0,n (µ) = moduli space of cone metrics on S 2 with given angles M 0,! M g, g = 2 X : y 1 =(x b 1 ) (x b 4 ) Theorem: M 0,n (µ) is naturally a complex hyperbolic manifold Z/1 acts on H 1 (X) [!] =[dx/y ] 2 H 1 (X) q (locally CH n 3, via periods of!) /1 q = 1 Schwarz, Picard, Deligne-Mostow, Thurston, 196 Math Olympiad (b i )! (positive line in C 1,2 ) = CH 2 Also get rep of braid group B4 U(1,2) (Burau)

3 What is the volume of moduli space? Theorem 1.2 The complex hyperbolic volume of moduli space satisfies vol(m 0,n,g µ )=C n 3 ( 1) P +1 ( P 3)! ( max 0, 1 ) B 1 µ i. B P i B P P { } P: partitions of {1,...,n} into blocks B. special cases (different methods): Parker, Sauter Cone manifolds Example: Glue together spherical polyhedra along faces in pairs. Thurston: The metric completion M 0,n (µ) is a CH n 3 cone manifold. Gauss-Bonnet (M): Acompactcone manifoldofdimensionn satisfies Ψ(x) dv(x) = χ(m σ )Θ σ. M[n] σ = Proof of volume formula GB Volume = 14 ( 1) P +1 ( P 3)! ( max 0, 1 ) B 1 µ i. B P i B P χ(m σ )Θ σ. σ { } (+,,,) contributes a stratum M0,4. no stratum unless P B µ i < 1. Proof of cone GB uses polyhedral GB... Theorem 2.1 (Allendoerfer Weil) The Euler characteristic of a compact Riemannian polyhedron M of dimension n satisfies n 1 ( 1) n χ (M) = Ψ(x) dv(x) + M[n] Riemann curvature tensor by Ψ(x) = 2 ω n bundle to A defined by Ψ(x, ξ) = Ψ r,f (x, ξ) = 1 2 n/2 n! 0 2f r Ψ r,f (x, ξ), 2 ω 2f ω n 2f 1 r=0 ɛ(i)ɛ(j) g i,j S n M[r] where 1 2 f (2f)!(r 2f)! dv(x) Ψ(x, ξ) dξ. N(x) R i1i 2j 1j 2 R in 1i nj n 1j n. intrinsic K ɛ(i)ɛ(j) γ i,j S r R i1i 2j 1j 2 R i2f 1 i 2f j 2f 1 j 2f Λ i2f+1 j 2f+1 (ξ) Λ irj r (ξ). extrinsic K 1943

4 ...which in turns comes from Weyl s tube formula T[r] M[r] Example: μ = (,,,,)/1 16 Galois (continued) M[n] /1 q = 1 /1 q = 1 Figure 1. The tube around a polyhedron in R N+1. Non-arithmetic lattices (DM) (1-14)/36 = 1/9 M 0,n (µ) orbifold = CH 2 / ( discrete) = ( 0 dense) in U(1, 2). (1-10)/36 = 2/9 M 0,n M 0,n (µ 0 ) cone manifold holonomy 0 Invariants Kappes-Möller The volume ratios (µ, µ 0 )= vol(m 0,n(µ 0 )) vol(m 0,n (µ)) are the same for all subgroups of finite index in Γ. The 16 nonarithmetic lattices arising from moduli spaces fall into 9 commensurability classes. Γ is a nonarithmetic lattice

5 Volume ratios: examples q q (p i ) {ρ(µ, ν)} / / / / /16 1 1/ /33, 4/ /33, 4/ / / / / /3, 4/ /3, 4/ /61, 4/ /61, 4/61 (p i) χ(p (µ)) χ(m(µ)) /9-1/ /3 1/ /64-1/ /12 / /16-1/ /4-1/ /16 1/ / 1/ / 1/ / / / / /3-1/ /434-41/ /32 / / / /1296-1/ /6 63/ /64-13/ /216-11/ / / /1296 1/ / / /10-13/ /10 / /64 / /4-1/ /2-1/ /432 / /4-1/ /4 -/ /9-1/ /12 1/ /24 1/ /9-1/ /2 -/ /12 1/ /6 1/ /6 1/ /1 -/ /12 1/ /4 1/ /3 1/ /2 1/ /12-11/ /64 3/ /32 3/ /16 3/12 Table 3. Euler characteristics of the 94 orbifolds M(µ) and their cone manifold covers P (µ), with(µ i)=(p i/q). 94 orbifolds q (p i) χ(p (µ)) χ(m(µ)) /2 13/ /20 1/ / / /12-13/ /100 1/ /10 3/ /0-1/ /2 1/ /100 13/ /4 / / / /432-23/ /216-23/ /4 1/ / 1/ /4 / /6 1/ /24 / /144-31/ /2-23/ /12 1/ /4 13/ /24 / /4 1/ /2 1/ /24 11/ /24 13/ /12 / /49 1/ /49 1/ / 3/ /10 13/ /2 2/ /4 13/ /10 13/ /2 13/ /2 2/ /400 33/ /200 23/ /200 99/ /96 11/ /24 11/ / 2/ /300 3/ / / / 61/ /14 61/2 Table 3. (continued) Coda: nonarithmetic Fuchsian groups μ = (3,3,,)/10 μ = (1,1,9,9)/10 CH 1 CH 1 F Γ Γ Figure 6. A holomorphic map fromb3 an ideal pentagon to an ideal star. /2 T0,4 ρ=1/3 Curves on 2 surfaces. Let f : V M 2 be a Teichmüller 2 curve generated by (X, ω 2 ). Assume that SL(X, ω) hasrealquadratictracefieldk R. Let k k denote the Galois involution of K/Q, sendinga + b d to a b d, and let g g denote the corresponding involution on SL 2 (K).

The Gauss Bonnet theorem for cone manifolds and volumes of moduli spaces

The Gauss Bonnet theorem for cone manifolds and volumes of moduli spaces Curtis T. McMullen 17 June 2013 Abstract This paper generalizes the Gauss Bonnet formula to a class of stratified spaces called