Moduli spaces and locally symmetric spaces

Transcription

1 p. 1/4 Moduli spaces and locally symmetric spaces Benson Farb University of Chicago Once it is possible to translate any particular proof from one theory to another, then the analogy has ceased to be productive for this purpose; it would cease to be at all productive if at one point we had a meaningful and natural way of deriving both theories from a single one.... Gone is the analogy: gone are the two theories, their conflicts and their delicious reciprocal reflections, their furtive caresses, their inexplicable quarrels; alas, all is just one theory, whose majestic beauty can no longer excite us. letter from André to Simone Weil, 1940 [stolen from Kent-Leininger]

2 p. 2/4 Prelude: A useful viewpoint Mod g := Homeo + (S g )/ Homeo 0 (S g ) Problem: Classify elements φ Mod g. Solution (Thurston, Bers): STEP 1. Look at action of Mod g on Teichmüller space T g.

3 p. 3/4 T 1 / Mod 1 = H 2 / SL(2,Z) [Picture stolen from C. McMullen] Reprise: Moduli space for genus g =

4 p. 4/4 Prelude: A useful viewpoint Mod g := Homeo + (S g )/ Homeo 0 (S g ) Problem: Classify elements φ Mod g. Solution (Thurston, Bers): STEP 1. Look at action of Mod g on Teichmüller space T g. Properly discontinuous By isometries (of Teichmüller metric)

5 p. 4/4 Prelude: A useful viewpoint Mod g := Homeo + (S g )/ Homeo 0 (S g ) Problem: Classify elements φ Mod g. Solution (Thurston, Bers): STEP 1. Look at action of Mod g on Teichmüller space T g. Properly discontinuous By isometries (of Teichmüller metric) STEP 2. Pretend T g is hyperbolic space H n. H n negatively curved (so d H n is convex) H n compactifies to a closed ball Simple dynamics on H n

6 p. 4/4 Prelude: A useful viewpoint Mod g := Homeo + (S g )/ Homeo 0 (S g ) Problem: Classify elements φ Mod g. Solution (Thurston, Bers): STEP 1. Look at action of Mod g on Teichmüller space T g. Properly discontinuous By isometries (of Teichmüller metric) STEP 2. Pretend T g is hyperbolic space H n. H n negatively curved (so d H n is convex) H n compactifies to a closed ball Simple dynamics on H n STEP 3. Two proof endings Bers: Analyze Minset(φ) Thurston: Apply Brouwer Fixed Point Theorem; analyze

7 p. 5/4 Prelude: The truth interferes T g is NOT negatively curved (unless g = 1)! Nontrivial problems 1. How to analyze Minset(φ)? 2. How to compactify T g?

8 p. 6/4 Prelude: T g vs. symmetric spaces How close is Teichmüller space to being a symmetric space? How much of the formal geometry of a symmetric space does Teichmüller space have? J. Harer, 1988 Some pioneers: Dehn, Bers, Thurston, Harvey, Harer, Ivanov, Masur, and many others.

9 p. 7/4 Talk Outline Theme: Think of moduli space M g := T g / Mod g as a locally symmetric orbifold. Today Use this philosophy to discover conjectures/theorems. Find universal constraints on this philosophy. 1. Symmetry and homogeneity 2. Reduction theory Other aspects Curvature (Riemannian, holomorphic, coarse) Convex cocompact groups Rank invariants (R-rank, Q-rank, geometric ranks) Rank one / higher rank dichotomy Compactifications

10 p. 8/4 Locally symmetric spaces: Examples Finite volume hyperbolic manifolds M = Γ\H n V g = SL(n,Z)\ SL(n,R)/ SO(n) = moduli space of flat, unit volume, n-dimensional tori A g = Sp(2g,Z)\ Sp(2g,R)/ U(g) = moduli space of g-dimensional, principally polarized abelian varieties M = Γ\G/K, G semisimple, K max compact, Γ lattice. Remarks M 1 = V 2 M g A g (Torelli Theorem)

11 p. 9/4 Locally symmetric spaces M: First Properties M finite volume Riemannian Symmetry: M is symmetric at every point: the flip map is an isometry. γ(t) γ( t) Homogeneity: Isom( M) acts transitively on M. Curvature: K(M) 0 Algebraic system: M = Γ\G/K Algebraic formulas for differential-geometric quantities Rigidity properties π 1 (M) usually determines M (Mostow, Prasad, Margulis)

12 p. 10/4 Dictionary: First entries Nonlinear Moduli space M g Linear loc. sym. space M Mod g π 1 (M) T g symmetric space M

13 Symmetry: Royden s Theorem p. 11/4

14 p. 12/4 Symmetry: Royden s Theorem Teichmüller metric d T Complete Finsler metric. Induced by norm on TX (T g) = QD(X): φ T := X φ(z) dz 2

15 p. 12/4 Symmetry: Royden s Theorem Teichmüller metric d T Complete Finsler metric. Induced by norm on TX (T g) = QD(X): φ T := Vol(M g, d T ) < (Masur) X φ(z) dz 2

16 p. 12/4 Symmetry: Royden s Theorem Teichmüller metric d T Complete Finsler metric. Induced by norm on TX (T g) = QD(X): φ T := Vol(M g, d T ) < (Masur) X φ(z) dz 2 Royden s Theorem (1971). Let g 3. Then

17 p. 12/4 Symmetry: Royden s Theorem Teichmüller metric d T Complete Finsler metric. Induced by norm on TX (T g) = QD(X): φ T := Vol(M g, d T ) < (Masur) X φ(z) dz 2 Royden s Theorem (1971). Let g 3. Then 1. There is no point X T g at which T g is symmetric.

18 p. 12/4 Symmetry: Royden s Theorem Teichmüller metric d T Complete Finsler metric. Induced by norm on TX (T g) = QD(X): φ T := Vol(M g, d T ) < (Masur) X φ(z) dz 2 Royden s Theorem (1971). Let g 3. Then 1. There is no point X T g at which T g is symmetric. 2. Isom(T g ) = Mod ± g.

19 Symmetry: Intrinsic inhomogeneity of T g p. 13/4

20 p. 14/4 Symmetry: Intrinsic inhomogeneity of T g Theorem (Farb-Weinberger). For any complete, Mod g -invariant Riemannian (or even Finsler) metric on T g with Vol(M g ) < :

21 p. 14/4 Symmetry: Intrinsic inhomogeneity of T g Theorem (Farb-Weinberger). For any complete, Mod g -invariant Riemannian (or even Finsler) metric on T g with Vol(M g ) < : 1. There is no point X T g at which T g is symmetric.

22 p. 14/4 Symmetry: Intrinsic inhomogeneity of T g Theorem (Farb-Weinberger). For any complete, Mod g -invariant Riemannian (or even Finsler) metric on T g with Vol(M g ) < : 1. There is no point X T g at which T g is symmetric. 2. Isom(T g ) is discrete and [Isom(T g ) : Mod g ] <.

23 p. 14/4 Symmetry: Intrinsic inhomogeneity of T g Theorem (Farb-Weinberger). For any complete, Mod g -invariant Riemannian (or even Finsler) metric on T g with Vol(M g ) < : 1. There is no point X T g at which T g is symmetric. 2. Isom(T g ) is discrete and [Isom(T g ) : Mod g ] <. Corollary (Ivanov): M g admits no locally symmetric metric.

24 p. 14/4 Symmetry: Intrinsic inhomogeneity of T g Theorem (Farb-Weinberger). For any complete, Mod g -invariant Riemannian (or even Finsler) metric on T g with Vol(M g ) < : 1. There is no point X T g at which T g is symmetric. 2. Isom(T g ) is discrete and [Isom(T g ) : Mod g ] <. Corollary (Ivanov): M g admits no locally symmetric metric. Conjecture: [Isom(T g ) : Mod g ] 2.

25 p. 14/4 Symmetry: Intrinsic inhomogeneity of T g Theorem (Farb-Weinberger). For any complete, Mod g -invariant Riemannian (or even Finsler) metric on T g with Vol(M g ) < : 1. There is no point X T g at which T g is symmetric. 2. Isom(T g ) is discrete and [Isom(T g ) : Mod g ] <. Corollary (Ivanov): M g admits no locally symmetric metric. Conjecture: [Isom(T g ) : Mod g ] 2. Stronger version: Theorem holds for finite covers of M g.

26 p. 14/4 Symmetry: Intrinsic inhomogeneity of T g Theorem (Farb-Weinberger). For any complete, Mod g -invariant Riemannian (or even Finsler) metric on T g with Vol(M g ) < : 1. There is no point X T g at which T g is symmetric. 2. Isom(T g ) is discrete and [Isom(T g ) : Mod g ] <. Corollary (Ivanov): M g admits no locally symmetric metric. Conjecture: [Isom(T g ) : Mod g ] 2. Stronger version: Theorem holds for finite covers of M g. Remark. Gives no info on WP metric. But Isom(T g, WP) = Mod ± g (Masur-Wolf).

27 Teichmüller geometry of moduli space p. 15/4

28 p. 16/4 Teichmüller geometry of moduli space Goals (with H. Masur) 1. Understand the geometry of M g with Teich metric.

29 p. 16/4 Teichmüller geometry of moduli space Goals (with H. Masur) 1. Understand the geometry of M g with Teich metric. 2. Build a Reduction Theory for (T g, Mod g ) (as with arithmetic groups)

30 p. 16/4 Teichmüller geometry of moduli space Goals (with H. Masur) 1. Understand the geometry of M g with Teich metric. 2. Build a Reduction Theory for (T g, Mod g ) (as with arithmetic groups) 3. Compactifications (à la Siegel, Borel, Ji, Macpherson, Leuzinger, etc.)

31 p. 16/4 Teichmüller geometry of moduli space Goals (with H. Masur) 1. Understand the geometry of M g with Teich metric. 2. Build a Reduction Theory for (T g, Mod g ) (as with arithmetic groups) 3. Compactifications (à la Siegel, Borel, Ji, Macpherson, Leuzinger, etc.)

32 Teich geometry of M g : Cone at infinity p. 17/4

33 p. 18/4 Teich geometry of M g : Cone at infinity DEFINITION. (M, d) pointed metric space. Cone(M) := lim n (M, 1 n d)

34 p. 19/4 Teich geometry of M g : Cone at infinity DEFINITION. (M, d) pointed metric space. Cone(M) := lim n (M, 1 n d) Example. Cone(M) for M finite volume hyperbolic.

35 Finite volume hyperbolic manifold M p. 20/4

36 1 M 10 p. 21/4

37 1 M 20 p. 22/4

38 1 M 40 p. 23/4

39 1 M p. 24/4

40 p. 25/4 Teich geometry of M g : Cone at infinity DEFINITION. (M, d) pointed metric space. Cone(M) := lim n (M, 1 n d) Example. Cone(M) for M finite volume hyperbolic.

41 p. 26/4 Teich geometry of M g : Cone at infinity DEFINITION. (M, d) pointed metric space. Cone(M) := lim n (M, 1 n d) Example. Cone(M) for M finite volume hyperbolic. Question. What is Cone(M g )?

42 Teich geometry of M g : Almost isometric model p. 27/4

43 p. 28/4 Teich geometry of M g : Almost isometric model Build metric simplicial complex V g

44 p. 29/4 Teich geometry of M g : Almost isometric model Build metric simplicial complex V g 1. C g = complex of curves on Σ g

45 p. 29/4 Teich geometry of M g : Almost isometric model Build metric simplicial complex V g 1. C g = complex of curves on Σ g 2. Let V g = [ [0, ) C g {0} C g ]/ Mod g

46 p. 29/4 Teich geometry of M g : Almost isometric model Build metric simplicial complex V g 1. C g = complex of curves on Σ g 2. Let V g = [ [0, ) C g {0} C g ]/ Mod g 3. Metrize the cone over each σ C g with sup metric.

47 p. 29/4 Teich geometry of M g : Almost isometric model Build metric simplicial complex V g 1. C g = complex of curves on Σ g 2. Let V g = [ [0, ) C g {0} C g ]/ Mod g 3. Metrize the cone over each σ C g with sup metric. 4. Endow V g with induced path metric.

48 The 2-punctured torus case: V 1,2 p. 30/4

49 p. 31/4 Teich geometry of M g : Almost isometric model Theorem (Farb-Masur). There is an injective map Ψ : V g M g which is an almost isometry:

50 p. 31/4 Teich geometry of M g : Almost isometric model Theorem (Farb-Masur). There is an injective map Ψ : V g M g which is an almost isometry: There exists D > 0 such that 1. For all x, y V g : d Vg (x, y) D d Mg (Ψ(x), Ψ(y)) d Vg (x, y) + D 2. Nbhd D (ψ(v g )) = M g

51 p. 31/4 Teich geometry of M g : Almost isometric model Theorem (Farb-Masur). There is an injective map Ψ : V g M g which is an almost isometry: There exists D > 0 such that 1. For all x, y V g : d Vg (x, y) D d Mg (Ψ(x), Ψ(y)) d Vg (x, y) + D 2. Nbhd D (ψ(v g )) = M g

52 p. 32/4 Teich geometry of M g : Almost isometric model Corollary. Cone(M g ) = V g. [Note POSITIVE CURVATURE!]

53 p. 32/4 Teich geometry of M g : Almost isometric model Corollary. Cone(M g ) = V g. [Note POSITIVE CURVATURE!] Application (Farb-Weinberger). Sharp results on (non)existence of PSC metrics on M g.

54 p. 32/4 Teich geometry of M g : Almost isometric model Corollary. Cone(M g ) = V g. [Note POSITIVE CURVATURE!] Application (Farb-Weinberger). Sharp results on (non)existence of PSC metrics on M g. Classical (Hattori). Cone(Γ\G/K) = Γ-quotient of top. cone on Tits building Q (G).

55 p. 32/4 Teich geometry of M g : Almost isometric model Corollary. Cone(M g ) = V g. [Note POSITIVE CURVATURE!] Application (Farb-Weinberger). Sharp results on (non)existence of PSC metrics on M g. Classical (Hattori). Cone(Γ\G/K) = Γ-quotient of top. cone on Tits building Q (G). Key ingredient in proof: Minsky Product Regions Theorem.

56 p. 33/4 Dictionary: Reduction theory Nonlinear Linear

57 p. 33/4 Dictionary: Reduction theory Nonlinear simple closed curve Linear vector

58 p. 33/4 Dictionary: Reduction theory Nonlinear simple closed curve r-tuple of disjoint curves Linear vector subspace of Q n

59 p. 33/4 Dictionary: Reduction theory Nonlinear simple closed curve r-tuple of disjoint curves bs(c g ) = flags of tuples Linear vector subspace of Q n Tits bldng Q = flgs of subspaces

60 p. 33/4 Dictionary: Reduction theory Nonlinear simple closed curve r-tuple of disjoint curves bs(c g ) = flags of tuples d(σ g,n ) = 3g 3 + n Linear vector subspace of Q n Tits bldng Q = flgs of subspaces Q-rank

61 p. 33/4 Dictionary: Reduction theory Nonlinear simple closed curve r-tuple of disjoint curves bs(c g ) = flags of tuples d(σ g,n ) = 3g 3 + n stabilizer of disjoint r-tuple Linear vector subspace of Q n Tits bldng Q = flgs of subspaces Q-rank parabolic subgroup (stab. of flag)

62 p. 33/4 Dictionary: Reduction theory Nonlinear simple closed curve r-tuple of disjoint curves bs(c g ) = flags of tuples d(σ g,n ) = 3g 3 + n stabilizer of disjoint r-tuple stab(α) Linear vector subspace of Q n Tits bldng Q = flgs of subspaces Q-rank parabolic subgroup (stab. of flag) max. parabolic 0 0

63 p. 33/4 Dictionary: Reduction theory Nonlinear simple closed curve r-tuple of disjoint curves bs(c g ) = flags of tuples d(σ g,n ) = 3g 3 + n stabilizer of disjoint r-tuple stab(α) stab(max curve syst.) Z 3g 3 Linear vector subspace of Q n Tits bldng Q = flgs of subspaces Q-rank parabolic subgroup (stab. of flag) max. parabolic 0 0 min. parabolic solvabl

64 p. 33/4 Dictionary: Reduction theory Nonlinear simple closed curve r-tuple of disjoint curves bs(c g ) = flags of tuples d(σ g,n ) = 3g 3 + n stabilizer of disjoint r-tuple stab(α) stab(max curve syst.) Z 3g 3 marked length inequalities Linear vector subspace of Q n Tits bldng Q = flgs of subspaces Q-rank parabolic subgroup (stab. of flag) max. parabolic 0 0 min. parabolic Weyl chamber solvabl

65 p. 34/4 Convex cocompactness: Teich geom. and extensions Farb-Mosher

66 p. 35/4 Convex cocompactness: Teich geom. and extensions Farb-Mosher 1 π 1 Σ g Γ G G 1

67 p. 36/4 Convex cocompactness: Teich geom. and extensions Farb-Mosher 1 π 1 Σ g Γ G G 1 Fact. Γ G is determined by ρ : G Out(π 1 Σ g ) Mod g

68 p. 37/4 Convex cocompactness: Teich geom. and extensions Farb-Mosher 1 π 1 Σ g Γ G G 1 Fact. Γ G is determined by ρ : G Out(π 1 Σ g ) Mod g Idea. { Group theory and geometry of Γ G } { Geometry of the ρ(g)-orbit in T g }

69 p. 38/4 Convex cocompactness: Definition and examples DEFINITION. A subgroup G < Mod g is convex cocompact if some (any) orbit G x T g is quasiconvex.

70 p. 39/4 Convex cocompactness: Definition and examples DEFINITION. A subgroup G < Mod g is convex cocompact if some (any) orbit G x T g is quasiconvex. Examples. (Thurston) G = Z is convex cocompact iff Γ G is δ-hyperbolic.

71 p. 39/4 Convex cocompactness: Definition and examples DEFINITION. A subgroup G < Mod g is convex cocompact if some (any) orbit G x T g is quasiconvex. Examples. (Thurston) G = Z is convex cocompact iff Γ G is δ-hyperbolic. (Genericity) For any {φ 1,...,φ r } pseudo-anosovs, N > 0 so that the group is convex cocompact. < {φ N 1,...,φ N r } >

72 p. 40/4 Convex cocompactness: Hyperbolic Extensions Conjecture Conjecture. The following are equivalent: 1. Γ G is δ-hyperbolic. 2. ker(ρ) is finite and ρ(g) is convex cocompact.

73 p. 41/4 Convex cocompactness: Hyperbolic Extensions Conjecture Conjecture. The following are equivalent: 1. Γ G is δ-hyperbolic. 2. ker(ρ) is finite and ρ(g) is convex cocompact. Progress. FM: True for G free Applied to prove commensurator and quasi-isometric rigidity theorems for Γ G.

74 p. 41/4 Convex cocompactness: Hyperbolic Extensions Conjecture Conjecture. The following are equivalent: 1. Γ G is δ-hyperbolic. 2. ker(ρ) is finite and ρ(g) is convex cocompact. Progress. FM: True for G free Applied to prove commensurator and quasi-isometric rigidity theorems for Γ G. FM: (1) implies (2).

75 p. 41/4 Convex cocompactness: Hyperbolic Extensions Conjecture Conjecture. The following are equivalent: 1. Γ G is δ-hyperbolic. 2. ker(ρ) is finite and ρ(g) is convex cocompact. Progress. FM: True for G free Applied to prove commensurator and quasi-isometric rigidity theorems for Γ G. FM: (1) implies (2). Kent, Leinininger, Schleimer: new tools, examples.

76 p. 41/4 Convex cocompactness: Hyperbolic Extensions Conjecture Conjecture. The following are equivalent: 1. Γ G is δ-hyperbolic. 2. ker(ρ) is finite and ρ(g) is convex cocompact. Progress. FM: True for G free Applied to prove commensurator and quasi-isometric rigidity theorems for Γ G. FM: (1) implies (2). Kent, Leinininger, Schleimer: new tools, examples. Hamenstadt: Conjecture is true!

77 p. 41/4 Convex cocompactness: Hyperbolic Extensions Conjecture Conjecture. The following are equivalent: 1. Γ G is δ-hyperbolic. 2. ker(ρ) is finite and ρ(g) is convex cocompact. Progress. FM: True for G free Applied to prove commensurator and quasi-isometric rigidity theorems for Γ G. FM: (1) implies (2). Kent, Leinininger, Schleimer: new tools, examples. Hamenstadt: Conjecture is true!

78 p. 41/4 Convex cocompactness: Hyperbolic Extensions Conjecture Conjecture. The following are equivalent: 1. Γ G is δ-hyperbolic. 2. ker(ρ) is finite and ρ(g) is convex cocompact. Progress. FM: True for G free Applied to prove commensurator and quasi-isometric rigidity theorems for Γ G. FM: (1) implies (2). Kent, Leinininger, Schleimer: new tools, examples. Hamenstadt: Conjecture is true! Open Question (Kapovich, Mess). Is there a bundle Σ g M 4 Σ h with π 1 (M) δ-hyperbolic? With K(M) < 0?

79 p. 41/4 Convex cocompactness: Hyperbolic Extensions Conjecture Conjecture. The following are equivalent: 1. Γ G is δ-hyperbolic. 2. ker(ρ) is finite and ρ(g) is convex cocompact. Progress. FM: True for G free Applied to prove commensurator and quasi-isometric rigidity theorems for Γ G. FM: (1) implies (2). Kent, Leinininger, Schleimer: new tools, examples. Hamenstadt: Conjecture is true! Open Question (Kapovich, Mess). Is there a bundle Σ g M 4 Σ h with π 1 (M) δ-hyperbolic? With K(M) < 0?