What is the Euler characteristic of moduli space ? Compactified - PowerPoint PPT Presentation

Illustration The Gauss-Bonnet theorem for cone manifolds F and volumes of moduli spaces Γ Γ′ Curtis McMullen Avg of |F ′ | 2 in the hyperbolic metric? Holomorphic Galois conjugation Answer: 1/3 Kappes--Möller (2012) Thurston (1998) 5 5/2 F Allendoerfer--Weil (1943) 1-1 ∞ 2 ∞ 2 orbifold cone manifold Schwarz, Picard, Deligne--Mostow, Cohen--Wolfart, Parker, Sauter,.... -1+1/2+1/5 = -3/10 -1+1/2+2/5 = -1/10 What is the Euler characteristic of moduli space ? Compactified moduli space ? n 3 4 5 6 7 C n : b i 6 = b j } / Aut b M 0 ,n = { ( b 1 , . . . , b n ) 2 b C χ ( M 0 ,n ) 1 -1 2 -6 24 manifold Σ 0 ,n − 1 → M 0 ,n 1 2 7 34 213 χ ( M 0 ,n ) Fibration χ ( Σ 0 ,n − 1 ) = − ( n − 3) M 0 ,n − 1 Example: n=5 χ ( M 0 ,n ) = ( − 1) n +1 ( n − 3)! χ ( P 1 × P 1 − 7 P 1 + 12 P 0 ) = 4 − 14 + 12 = 2 Theorem M 0 ,n 6 = CH n − 3 / Γ for n>4. χ ( P 1 × P 1 + 3 P 1 − 3 P 0 ) = 4 + 6 − 3 = 7

Generating functions Moduli spaces of polyhedra... Fix µ 1 , . . . , µ n , 0 < µ i < 1 , P µ i = 2. Corollary 2 (Getzler) The generating functions Any ( b i ) determines a meromorphic 1-form on b C : ∞ χ ( M 0 ,n +1 ) x n ∞ χ ( M 0 ,n +1 ) x n � � f ( x ) = x − g ( x ) = x + and dx n ! n ! X ω = ( ω ) = µ i b i Q n n =2 n =2 1 ( x − b i ) µ i divisor of degree 2 are formal inverses of one another. ( b = convex polyhedron in R 3 C , | ω | ) ∼ Universal; via stable trees (M, L’Ens. math.) Cone angles 2 π (1 − µ i ). Example: n=12, μ i = 1/6 n=4 n=5 Example: μ = (7,7,7,7,8)/18 ... are complex hyperbolic after all 8 M 0 , 5 → M g , g = 25 M 0 ,n ( µ ) = moduli space of cone metrics on S 2 with given angles X : y 18 = ( x − b 1 ) · · · ( x − b 4 ) 7 7 Z / 18 acts on H 1 ( X ) 7 7 Theorem: M 0 ,n ( µ ) is naturally a complex hyperbolic manifold [ ω ] = [ dx/y 7 ] ∈ H 1 ( X ) q 7/18 (locally CH n − 3 , via periods of ω ) q = ζ − 7 18 ( b i ) 7! (positive line in C 1 , 2 ) ⇠ = CH 2 Schwarz, Picard, Deligne-Mostow, Thurston, 1986 Math Olympiad Also get rep of braid group B 4 → U(1,2) Signature (1,2) (Burau)

What is the volume of moduli space ? Cone manifolds § Example: Glue together spherical polyhedra along Theorem 1.2 The complex hyperbolic volume of moduli space satisfies faces in pairs. � | B | − 1 � � � � ( − 1) |P| +1 ( |P| − 3)! vol( M 0 ,n , g µ ) = C n − 3 max 0 , 1 − µ i . P B ∈ P i ∈ B Thurston: P : partitions of { 1 , . . . , n } into blocks B . The metric completion M 0 ,n ( µ ) is a CH n − 3 cone manifold. special cases (different methods): Parker, Sauter Gauss-Bonnet (M): .1 A compact cone–manifold of dimension n satisfies � � Ψ ( x ) dv ( x ) = χ ( M σ ) Θ σ . M [ n ] σ Proof of volume formula Proof of cone GB uses polyhedral GB... 1943 GB Theorem 2.1 (Allendoerfer–Weil) The Euler characteristic of a com- � Volume = ) = χ ( M σ ) Θ σ . pact Riemannian polyhedron M of dimension n satisfies σ n − 1 � � � ( − 1) n χ ′ ( M ) = � Ψ ( x ) dv ( x ) + dv ( x ) N ( x ) ∗ Ψ ( x, ξ ) d ξ . � | B | − 1 � M [ n ] M [ r ] r =0 = � ( − 1) |P| +1 ( |P| − 3)! � � max 0 , 1 − µ i . Riemann curvature tensor by 3 P B ∈ P i ∈ B � Ψ ( x ) = 2 1 � ( i ) � ( j ) � � � · R i 1 i 2 j 1 j 2 · · · R i n − 1 i n j n − 1 j n . 2 n/ 2 n ! g ω n i,j ∈ S n 8 intrinsic K bundle to A defined by 7 7 (7+7,7,7,8) contributes a � Ψ ( x, ξ ) = Ψ r,f ( x, ξ ) , where 0 ≤ 2 f ≤ r stratum ≃ M 0,4. 14 2 1 � ( i ) � ( j ) 7 7 � Ψ r,f ( x, ξ ) = · 2 f (2 f )!( r − 2 f )! · × ω 2 f ω n − 2 f − 1 γ i,j ∈ S r no stratum unless P B µ i < 1. R i 1 i 2 j 1 j 2 · · · R i 2 f − 1 i 2 f j 2 f − 1 j 2 f Λ i 2 f +1 j 2 f +1 ( ξ ) · · · Λ i r j r ( ξ ) . extrinsic K

Example: μ = (7,7,7,7,8)/18 (continued) ...which in turns comes from Weyl’s tube formula. → 1939 16 8 Galois 5 5 T[r] 7 7 M[r] 5 5 7 7 M[n] 5/18 q = ζ − 5 18 7/18 q = ζ − 7 18 Figure 1. The tube around a polyhedron in R N +1 . Signature (1,2) Signature (1,2) Non-arithmetic lattices (DM) Invariants 16 8 Kappes-Möller 5 7 5 7 ρ ( µ, µ 0 ) = vol( M 0 ,n ( µ 0 )) 10 The volume ratios 14 5 5 7 7 vol( M 0 ,n ( µ )) (18-10)/36 = 2/9 (18-14)/36 = 1/9 are the same for all subgroups of finite index in Γ . M 0 ,n M 0 ,n ( µ 0 ) M 0 ,n ( µ ) cone manifold The 16 nonarithmetic lattices arising orbifold from moduli spaces holonomy Γ 0 = CH 2 / Γ ∼ fall into 9 commensurability classes. = ( Γ 0 dense) in U (1 , 2). ( Γ discrete) ∼ Γ is a nonarithmetic lattice

Volume ratios: examples q ( p i ) χ ( P ( µ )) χ ( M ( µ )) 94 orbifolds ( p i ) χ ( P ( µ )) χ ( M ( µ )) q 3 1 1 1 1 1 1 -4/9 -1/1620 9 4 4 4 4 2 13/27 13/648 3 2 1 1 1 1 1/3 1/72 10 7 4 4 4 1 3/20 1/40 4 1 1 1 1 1 1 1 1 -15/64 -1/172032 10 3 3 3 3 3 3 2 293/1000 293/720000 q ( p i ) { ρ ( µ, ν ) } 4 2 1 1 1 1 1 1 25/128 5/18432 10 6 3 3 3 3 2 -26/125 -13/1500 4 3 1 1 1 1 1 -1/16 -1/1920 10 9 3 3 3 2 3/100 1/200 4 2 2 1 1 1 1 -1/4 -1/192 10 6 6 3 3 2 3/10 3/40 12 7 5 3 3 3 3 1/93 4 3 2 1 1 1 3/16 1/32 10 5 3 3 3 3 3 -17/50 -17/6000 4 2 2 2 1 1 3/8 1/32 10 8 3 3 3 3 3/25 1/200 5 2 2 2 2 2 3/5 1/200 10 6 5 3 3 3 39/100 13/200 12 8 7 3 3 3 1/13 6 1 1 1 1 1 1 1 1 1 1 1 1 -28315/419904 -809/5746705367040 12 8 5 5 5 1 7/48 7/288 6 2 1 1 1 1 1 1 1 1 1 1 5663/93312 809/48372940800 12 7 7 2 2 2 2 2 575/10368 115/497664 6 3 1 1 1 1 1 1 1 1 1 -119/3888 -17/201553920 12 9 7 2 2 2 2 -23/432 -23/10368 12 6 5 5 4 4 1/13 6 2 2 1 1 1 1 1 1 1 1 -287/4374 -41/50388480 12 7 7 4 2 2 2 -23/216 -23/2592 6 4 1 1 1 1 1 1 1 1 49/5832 7/33592320 12 11 7 2 2 2 1/48 1/288 6 3 2 1 1 1 1 1 1 1 2107/46656 301/33592320 12 9 9 2 2 2 1/8 1/96 12 7 6 5 3 3 1/17 6 5 1 1 1 1 1 1 1 -1/1296 -1/6531840 12 9 7 4 2 2 7/48 7/96 6 2 2 2 1 1 1 1 1 1 637/7776 637/33592320 12 7 7 6 2 2 1/6 1/24 6 4 2 1 1 1 1 1 1 -13/648 -13/466560 18 13 7 7 7 2 1/16 12 7 7 4 4 2 7/24 7/96 6 3 3 1 1 1 1 1 1 -11/216 -11/311040 12 7 5 3 3 3 3 -31/144 -31/3456 6 3 2 2 1 1 1 1 1 -91/1296 -91/311040 12 5 5 5 3 3 3 -23/72 -23/2592 6 5 2 1 1 1 1 1 5/1296 1/31104 18 8 7 7 7 7 1/16 12 10 5 3 3 3 1/12 1/72 6 4 3 1 1 1 1 1 55/1296 11/31104 12 8 7 3 3 3 13/48 13/288 6 2 2 2 2 1 1 1 1 -13/108 -13/62208 12 8 5 5 3 3 7/24 7/96 6 4 2 2 1 1 1 1 5/108 5/5184 20 14 11 5 5 5 1/33, 4/33 12 7 6 5 3 3 17/48 17/96 6 3 3 2 1 1 1 1 55/648 55/31104 12 6 5 5 5 3 1/2 1/12 6 5 3 1 1 1 1 -1/54 -1/1296 12 7 5 4 4 4 11/24 11/144 6 4 4 1 1 1 1 -2/27 -1/648 20 10 9 9 6 6 1/33, 4/33 12 6 5 5 4 4 13/24 13/96 6 3 2 2 2 1 1 1 55/432 55/15552 12 5 5 5 5 4 7/12 7/288 6 5 2 2 1 1 1 -1/54 -1/648 14 11 5 5 5 2 6/49 1/49 6 4 3 2 1 1 1 -5/54 -5/324 20 13 9 6 6 6 1/46 14 8 5 5 5 5 24/49 1/49 6 3 3 3 1 1 1 -1/9 -1/324 15 8 6 6 6 4 37/75 37/450 6 5 4 1 1 1 1/12 1/72 6 2 2 2 2 2 1 1 5/24 1/1152 18 11 8 8 8 1 13/108 13/648 12 7 5 4 4 4 1/22 18 13 7 7 7 2 4/27 2/81 6 4 2 2 2 1 1 -1/9 -1/108 18 10 10 7 7 2 13/54 13/216 6 3 3 2 2 1 1 -5/27 -5/216 6 5 3 2 1 1 1/12 1/24 18 14 13 3 3 3 13/108 13/648 24 19 17 4 4 4 1/22 18 10 7 7 7 5 13/27 13/162 6 4 4 2 1 1 1/6 1/24 6 4 3 3 1 1 1/6 1/24 18 8 7 7 7 7 16/27 2/81 6 3 2 2 2 2 1 -5/18 -5/432 20 14 11 5 5 5 99/400 33/800 24 14 9 9 9 7 1/22 6 5 2 2 2 1 1/12 1/72 20 13 9 6 6 6 69/200 23/400 6 4 3 2 2 1 1/4 1/8 20 10 9 9 6 6 99/200 99/800 6 3 3 3 2 1 1/3 1/18 24 19 17 4 4 4 11/96 11/576 15 8 6 6 6 4 1/37, 4/37 6 3 3 2 2 2 1/2 1/24 24 14 9 9 9 7 11/24 11/144 8 3 3 3 3 3 1 -33/128 -11/5120 30 26 19 5 5 5 4/75 2/225 8 6 3 3 3 1 9/64 3/128 30 23 22 5 5 5 37/300 37/1800 30 23 22 5 5 5 1/37, 4/37 8 5 5 2 2 2 9/32 3/128 30 22 11 9 9 9 16/75 8/225 8 4 3 3 3 3 9/16 3/128 42 34 29 7 7 7 61/588 61/3528 42 26 15 15 15 13 61/147 61/882 42 34 29 7 7 7 1/61, 4/61 Table 3. (continued) Table 3. Euler characteristics of the 94 orbifolds M ( µ ) and their 42 26 15 15 15 13 1/61, 4/61 cone manifold covers P ( µ ) , with ( µ i ) = ( p i /q ) . Coda: nonarithmetic μ = (3,3,7,7)/10 Fuchsian groups μ′ = (1,1,9,9)/10 ⊂ CH 1 CH 1 F Γ Γ′ ρ =1/3 B 3 5 5/2 T 0,4 ∞ ∞ 2 2