Filling invariants for lattices in symmetric spaces

Transcription

1 Filling invariants for lattices in symmetric spaces Robert Young (joint work with Enrico Leuzinger) New York University September 2016 This work was partly supported by a Sloan Research Fellowship, by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada, and by NSF grant DMS

2 Conjecture (Thurston, Gromov, Leuzinger-Pittet, Bestvina-Eskin-Wortman) In a nonuniform lattice in a rank-k symmetric space, spheres with dimension k 2 have polynomial filling volume, but there are (k 1)-dimensional spheres with exponential filling volume.

3 Conjecture (Thurston, Gromov, Leuzinger-Pittet, Bestvina-Eskin-Wortman) In a nonuniform lattice in a rank-k symmetric space, spheres with dimension k 2 have polynomial filling volume, but there are (k 1)-dimensional spheres with exponential filling volume. Theorem (Leuzinger-Y.) If Γ is a nonuniform lattice in a symmetric space of rank k 2 and n < k, then FV n Γ (V ) V n n 1 FV k 1 Γ (V ) exp(v k 1 ).

4 Filling invariants: Measuring connectivity Let X be an (n 1)-connected simplicial complex or manifold and let α C n 1 (X ) be a cycle. Define FV n (α) = inf β C n(x ) β=α mass β.

5 Filling invariants: Measuring connectivity Let X be an (n 1)-connected simplicial complex or manifold and let α C n 1 (X ) be a cycle. Define FV n (α) = FV n X (V ) = inf β C n(x ) β=α sup α C n 1 (X ) mass(α) V mass β. FV n (α).

6 Filling invariants: Measuring connectivity Let X be an (n 1)-connected simplicial complex or manifold and let α C n 1 (X ) be a cycle. Define FV n (α) = FV n X (V ) = inf β C n(x ) β=α sup α C n 1 (X ) mass(α) V mass β. FV n (α). FV 2 X (n) is also known as the homological Dehn function

7 Filling invariants: Measuring connectivity Let X be an (n 1)-connected simplicial complex or manifold and let α C n 1 (X ) be a cycle. Define FV n (α) = FV n X (V ) = inf β C n(x ) β=α sup α C n 1 (X ) mass(α) V mass β. FV n (α). FV 2 X (n) is also known as the homological Dehn function FV 2 R 2 (2πr) = πr 2

8 Filling invariants: Measuring connectivity Let X be an (n 1)-connected simplicial complex or manifold and let α C n 1 (X ) be a cycle. Define FV n (α) = FV n X (V ) = inf β C n(x ) β=α sup α C n 1 (X ) mass(α) V mass β. FV n (α). FV 2 X (n) is also known as the homological Dehn function FV 2 R (2πr) = πr 2 2 FV n R k (r n 1 ) = C n r n for k n (i.e., FV n R k (V ) = C n V n n 1 )

9 Filling invariants as geometric group invariants If X and Y are bilipschitz equivalent, then there is a C > 0 such that FV n X (C 1 V ) FV n Y (V ) FVn X (CV ).

10 Filling invariants as geometric group invariants If X and Y are bilipschitz equivalent, then there is a C > 0 such that FV n X (C 1 V ) FV n Y (V ) FVn X (CV ). Theorem (Gromov, Epstein-Cannon-Holt-Levy-Paterson-Thurston) If X and Y are quasi-isometric and are, for instance, manifolds with bounded curvature or simplicial complexes with bounded degree, then FV n X and FVn Y are the same up to constants.

11 Filling invariants as geometric group invariants If X and Y are bilipschitz equivalent, then there is a C > 0 such that FV n X (C 1 V ) FV n Y (V ) FVn X (CV ). Theorem (Gromov, Epstein-Cannon-Holt-Levy-Paterson-Thurston) If X and Y are quasi-isometric and are, for instance, manifolds with bounded curvature or simplicial complexes with bounded degree, then FV n X and FVn Y are the same up to constants. In particular, if G is a group acting geometrically on an n connected space X, we can define FV n G = FVn X (up to constants).

12 Examples: negative curvature Small FV 2 is equivalent to negative curvature. If X has pinched negative curvature, then we can fill curves using geodesics. These discs have area linear in the length of their boundary, so FV 2 (n) n.

13 Examples: negative curvature Small FV 2 is equivalent to negative curvature. If X has pinched negative curvature, then we can fill curves using geodesics. These discs have area linear in the length of their boundary, so FV 2 (n) n. In fact, G is a group with sub-quadratic Dehn function (FV 2 n 2 ) if and only if G is δ-hyperbolic (Gromov).

14 Examples: nonpositive curvature and quadratic bounds Nonpositive curvature implies quadratic Dehn function: If X has nonpositive curvature, we can fill curves with geodesics, but the discs may have quadratically large area.

15 Examples: nonpositive curvature and quadratic bounds Nonpositive curvature implies quadratic Dehn function: If X has nonpositive curvature, we can fill curves with geodesics, but the discs may have quadratically large area. But the class of groups with quadratic Dehn functions is extremely rich; it includes Thompson s group (Guba), many solvable groups (Leuzinger-Pittet, de Cornulier-Tessera), some nilpotent groups (Gromov, Sapir-Ol shanskii, others), lattices in symmetric spaces (Druţu, Y., Cohen, others), and many more.

16 Examples: higher dimensions (Lang, Bonk-Schramm) If G is δ-hyperbolic, then FV n X (V ) V for all n.

17 Examples: higher dimensions (Lang, Bonk-Schramm) If G is δ-hyperbolic, then FV n X (V ) V for all n. (Gromov, Wenger) If X is complete and nonpositively curved, then FV n n X (V ) V n 1 for all n.

18 Examples: higher dimensions (Lang, Bonk-Schramm) If G is δ-hyperbolic, then FV n X (V ) V for all n. (Gromov, Wenger) If X is complete and nonpositively curved, then FV n n X (V ) V n 1 for all n. But subsets of nonpositively curved spaces can have stranger behavior!

19 Sol 3 and Sol 5 e t 0 x Sol 3 = 0 e t y x, y, t R 0 0 1

20 Sol 3 and Sol 5 e t 0 x Sol 3 = 0 e t y x, y, t R e t x Sol 5 = 0 e t 2 0 y 0 0 e t 3 z ti =

21 Sol 3 and Sol 5 e t 0 x Sol 3 = 0 e t y x, y, t R has FV 2 e n. (Gromov) e t x Sol 5 = 0 e t 2 0 y 0 0 e t 3 z ti = has FV 2 n 2. (Gromov, Leuzinger-Pittet)

22 Sol 3 and Sol 5 e a 0 x Sol 3 0 e b y {( )} {( )} e a x e b y = = H 2 H

23 Sol 3 and Sol 5 e a 0 x Sol 3 0 e b y {( )} {( )} e a x e b y = = H 2 H e a 0 0 x Sol 5 0 e b 0 y 0 0 e c z = H 2 H 2 H

24 Sol 3 and Sol 5 e a 0 x Sol 3 0 e b y {( )} {( )} e a x e b y = = H 2 H e a 0 0 x Sol 5 0 e b 0 y 0 0 e c z = H 2 H 2 H But Sol 5 has spheres which are exponentially difficult to fill!

25 Larger ranks In general, Sol 2k 1 (H 2 ) k

26 Larger ranks In general, Sol 2k 1 (H 2 ) k i.e., Sol 2k 1 is a subset of a symmetric space of rank k

27 Larger ranks In general, Sol 2k 1 (H 2 ) k i.e., Sol 2k 1 is a subset of a symmetric space of rank k So Sol 2k 1 contains (k 1)-spheres (intersections with flats) with exponentially large filling area (Gromov)

28 Larger ranks In general, Sol 2k 1 (H 2 ) k i.e., Sol 2k 1 is a subset of a symmetric space of rank k So Sol 2k 1 contains (k 1)-spheres (intersections with flats) with exponentially large filling area (Gromov) But there are plenty of lower-dimensional surfaces to fill lower-dimensional spheres, so FV n (V ) V n n 1 when n < k (Y.)

29 The main theorem Theorem (Leuzinger-Y.) If Γ is a nonuniform lattice in a symmetric space X of rank k 2 and n < k, then FV n Γ (V ) V n n 1 FV k 1 Γ (V ) exp(v k 1 ).

30 The main theorem Theorem (Leuzinger-Y.) If Γ is a nonuniform lattice in a symmetric space X of rank k 2 and n < k, then FV n Γ (V ) V n n 1 FV k 1 Γ (V ) exp(v k 1 ). A lattice in a symmetric space is a group that acts on the space with a quotient of finite volume

31 The main theorem Theorem (Leuzinger-Y.) If Γ is a nonuniform lattice in a symmetric space X of rank k 2 and n < k, then FV n Γ (V ) V n n 1 FV k 1 Γ (V ) exp(v k 1 ). A lattice in a symmetric space is a group that acts on the space with a quotient of finite volume When rank X 2, all lattices come from arithmetic constructions

32 The main theorem Theorem (Leuzinger-Y.) If Γ is a nonuniform lattice in a symmetric space X of rank k 2 and n < k, then FV n Γ (V ) V n n 1 FV k 1 Γ (V ) exp(v k 1 ). A lattice in a symmetric space is a group that acts on the space with a quotient of finite volume When rank X 2, all lattices come from arithmetic constructions, e.g.: SL n (Z) acting on the symmetric space SL n (R)/SO(n)

33 The main theorem Theorem (Leuzinger-Y.) If Γ is a nonuniform lattice in a symmetric space X of rank k 2 and n < k, then FV n Γ (V ) V n n 1 FV k 1 Γ (V ) exp(v k 1 ). A lattice in a symmetric space is a group that acts on the space with a quotient of finite volume When rank X 2, all lattices come from arithmetic constructions, e.g.: SL n (Z) acting on the symmetric space SL n (R)/SO(n) SL 2 (Z[ 2]) acting on H 2 H 2 (a Hilbert modular group)

34 The main theorem Theorem (Leuzinger-Y.) If Γ is a nonuniform lattice in a symmetric space X of rank k 2 and n < k, then FV n Γ (V ) V n n 1 FV k 1 Γ (V ) exp(v k 1 ). A lattice in a symmetric space is a group that acts on the space with a quotient of finite volume When rank X 2, all lattices come from arithmetic constructions, e.g.: SL n (Z) acting on the symmetric space SL n (R)/SO(n) SL 2 (Z[ 2]) acting on H 2 H 2 (a Hilbert modular group) A nonuniform lattice is a lattice that acts with noncompact quotient

35 Lattices act on subsets of X If Γ is a nonuniform lattice, the quotient Γ\X has cusps.

36 Lattices act on subsets of X If Γ is a nonuniform lattice, the quotient Γ\X has cusps. Cutting out the cusps corresponds to cutting out horoballs in X. Lemma If Γ is a nonuniform lattice, then there is an r 0 such that for r r 0, Γ acts geometrically on a set X (r) X such that X (r) is contractible and approximates the r neighborhood of Γ. We can write X (r) = X \ i H i, where the H i are a collection of horoballs in X.

37 Low dimensions Dimension 1: (Lubotzky-Mozes-Raghunathan) If X has rank 2, then d Γ (x, y) d X (r0 )(x, y) d G (x, y) for all x, y Γ.

38 Low dimensions Dimension 1: (Lubotzky-Mozes-Raghunathan) If X has rank 2, then d Γ (x, y) d X (r0 )(x, y) d G (x, y) for all x, y Γ. Dimension 2: (Leuzinger-Pittet) If Γ is an irreducible lattice in a symmetric space G of rank 2, then it has exponential Dehn function.

39 Low dimensions Dimension 1: (Lubotzky-Mozes-Raghunathan) If X has rank 2, then d Γ (x, y) d X (r0 )(x, y) d G (x, y) for all x, y Γ. Dimension 2: (Leuzinger-Pittet) If Γ is an irreducible lattice in a symmetric space G of rank 2, then it has exponential Dehn function. (Druţu) If Γ is an irreducible lattice of Q-rank 1 in a symmetric space X of rank 3, then FV 2 Γ (n) n2.

40 Low dimensions Dimension 1: (Lubotzky-Mozes-Raghunathan) If X has rank 2, then d Γ (x, y) d X (r0 )(x, y) d G (x, y) for all x, y Γ. Dimension 2: (Leuzinger-Pittet) If Γ is an irreducible lattice in a symmetric space G of rank 2, then it has exponential Dehn function. (Druţu) If Γ is an irreducible lattice of Q-rank 1 in a symmetric space X of rank 3, then FV 2 Γ (n) n2. (Y.) FV 2 SL p(z) (n) n2 when p 5 (i.e., rank 4).

41 Low dimensions Dimension 1: (Lubotzky-Mozes-Raghunathan) If X has rank 2, then d Γ (x, y) d X (r0 )(x, y) d G (x, y) for all x, y Γ. Dimension 2: (Leuzinger-Pittet) If Γ is an irreducible lattice in a symmetric space G of rank 2, then it has exponential Dehn function. (Druţu) If Γ is an irreducible lattice of Q-rank 1 in a symmetric space X of rank 3, then FV 2 Γ (n) n2. (Y.) FV 2 SL p(z) (n) n2 when p 5 (i.e., rank 4). (Cohen) FV 2 SP p(z) (n) n2 when p 5 (i.e., rank 5).

42 Higher dimensions Dimension > 2: (Epstein-Cannon-Holt-Levy-Paterson-Thurston) If Γ = SL k+1 (Z), then FV k Γ (r k 1 ) exp r.

43 Higher dimensions Dimension > 2: (Epstein-Cannon-Holt-Levy-Paterson-Thurston) If Γ = SL k+1 (Z), then FV k Γ (r k 1 ) exp r. (Wortman) If Γ is an irreducible lattice in a semisimple group G of rank k and its relative root system is not G 2, F 4, E 8, or BC n, then FV k Γ (r k 1 ) exp r.

44 Higher dimensions Dimension > 2: (Epstein-Cannon-Holt-Levy-Paterson-Thurston) If Γ = SL k+1 (Z), then FV k Γ (r k 1 ) exp r. (Wortman) If Γ is an irreducible lattice in a semisimple group G of rank k and its relative root system is not G 2, F 4, E 8, or BC n, then FV k Γ (r k 1 ) exp r. (Bestvina-Eskin-Wortman) If Γ is an irreducible lattice in a semisimple group G which is a product of n simple groups, then FV k Γ is bounded by a polynomial for k < n.

45 Higher dimensions Dimension > 2: (Epstein-Cannon-Holt-Levy-Paterson-Thurston) If Γ = SL k+1 (Z), then FV k Γ (r k 1 ) exp r. (Wortman) If Γ is an irreducible lattice in a semisimple group G of rank k and its relative root system is not G 2, F 4, E 8, or BC n, then FV k Γ (r k 1 ) exp r. (Bestvina-Eskin-Wortman) If Γ is an irreducible lattice in a semisimple group G which is a product of n simple groups, then FV k Γ is bounded by a polynomial for k < n. (Leuzinger-Y.) If Γ is an irreducible lattice of Q-rank 1 in a symmetric space X of rank k, then FV n Γ (r n 1 ) r n for n < k.

46 A flat in SL 3 (R)

47 A flat in SL 3 (R)

48 Lower bounds using random flats Results of Kleinbock and Margulis imply: Lemma (see Kleinbock-Margulis) There is a c > 1 such that if x X and ρ = d(x, Γ), then there is a flat E passing through x such that the sphere S E (cρ) E of radius cρ satisfies S E (x, cρ) X (c log ρ + c).

49 Lower bounds using random flats Results of Kleinbock and Margulis imply: Lemma (see Kleinbock-Margulis) There is a c > 1 such that if x X and ρ = d(x, Γ), then there is a flat E passing through x such that the sphere S E (cρ) E of radius cρ satisfies S E (x, cρ) X (c log ρ + c). This sphere has filling volume e ρ,

50 Lower bounds using random flats Results of Kleinbock and Margulis imply: Lemma (see Kleinbock-Margulis) There is a c > 1 such that if x X and ρ = d(x, Γ), then there is a flat E passing through x such that the sphere S E (cρ) E of radius cρ satisfies S E (x, cρ) X (c log ρ + c). This sphere has filling volume e ρ, and it can be retracted to a sphere that lies in X (r 0 ) at a cost of increasing the area by exp(c log ρ + c) ρ c.

51 Lower bounds using random flats Results of Kleinbock and Margulis imply: Lemma (see Kleinbock-Margulis) There is a c > 1 such that if x X and ρ = d(x, Γ), then there is a flat E passing through x such that the sphere S E (cρ) E of radius cρ satisfies S E (x, cρ) X (c log ρ + c). This sphere has filling volume e ρ, and it can be retracted to a sphere that lies in X (r 0 ) at a cost of increasing the area by exp(c log ρ + c) ρ c. Corollary FV k Γ (ρk 1+c ) e ρ.

52 Upper bounds Lemma (Y.) Since dim AN X <, we can prove upper bounds on FV n Γ by constructing a collection of simplices with vertices in Γ.

53 Upper bounds Lemma (Y.) Since dim AN X <, we can prove upper bounds on FV n Γ by constructing a collection of simplices with vertices in Γ. Sketch of proof If α : S n 1 X is a sphere, it has a filling β with mass β = FV n (α).

54 Upper bounds Lemma (Y.) Since dim AN X <, we can prove upper bounds on FV n Γ by constructing a collection of simplices with vertices in Γ. Sketch of proof If α : S n 1 X is a sphere, it has a filling β with mass β = FV n (α). By results of Lang and Schlichenmaier, we can triangulate β efficiently.

55 Upper bounds Lemma (Y.) Since dim AN X <, we can prove upper bounds on FV n Γ by constructing a collection of simplices with vertices in Γ. Sketch of proof If α : S n 1 X is a sphere, it has a filling β with mass β = FV n (α). By results of Lang and Schlichenmaier, we can triangulate β efficiently. We can use a triangulation of β as a template for assembling the simplices.

56 Upper bounds Lemma (Y.) Since dim AN X <, we can prove upper bounds on FV n Γ by constructing a collection of simplices with vertices in Γ. Sketch of proof If α : S n 1 X is a sphere, it has a filling β with mass β = FV n (α). By results of Lang and Schlichenmaier, we can triangulate β efficiently. We can use a triangulation of β as a template for assembling the simplices. How do we construct random simplices?

57 Filling using random flats Lemma (see Kleinbock-Margulis) There is a c > 1 such that if x, y Γ, ρ = d(x, y), and m is the midpoint of x and y, then there is a flat E passing through m such that d(x, E) < 1, d(y, E) < 1, and E \ B(m, cρ) is equidistributed in X. For example, for all R > cρ, E (B(m, R) \ B(m, cρ)) X (c + c log R log log R).

58 Filling using random flats Lemma (see Kleinbock-Margulis) There is a c > 1 such that if x, y Γ, ρ = d(x, y), and m is the midpoint of x and y, then there is a flat E passing through m such that d(x, E) < 1, d(y, E) < 1, and E \ B(m, cρ) is equidistributed in X. For example, for all R > cρ, E (B(m, R) \ B(m, cρ)) X (c + c log R log log R). Corollary If α : S n 1 X (r 0 ) and V = mass α, FV n n X (c+c log V )(α) V n 1. Using the retraction X (c + c log V ) X (r 0 ), FV n n c+ Γ (V ) V n 1.

59 Bootstrapping A filling is made of random flats:

60 Bootstrapping A filling is made of random flats: Cut out the parts of E that lie in a thin part.

61 Bootstrapping A filling is made of random flats: Cut out the parts of E that lie in a thin part. Replace them with a disc of polynomial area.

62 Bootstrapping A filling is made of random flats: Cut out the parts of E that lie in a thin part. Replace them with a disc of polynomial area. The result is a filling that lies in X (r 0 ) and has volume V n n 1.