Affine structures for closed 3-dimensional manifolds with Sol-geom

Transcription

1 Affine structures for closed 3-dimensional manifolds with Sol-geometry Jong Bum Lee 1 (joint with Ku Yong Ha) 1 Sogang University, Seoul, KOREA International Conference on Algebraic and Geometric Topology Capital Normal University, Beijing, China June 29, 2010

2 Thanks Thanks much organizers: Haibao Duan, Fuquan Fang, Jie Wu, Xuezhi Zhao for a kind invitation.

3 Motivation According to Thurston, there are 8 kinds of geometries in dimension 3. R 3, H 3, S 3, S 2 R, H 2 R, SL(2, R), Nil, Sol. A question naturally arisen is the problem of the classification of closed 3-manifolds with a geometric structure modeled on one of these eight types. 1 The 3-dim Euclidean space forms problem: the classification problem of closed 3-manifolds with R 3 -geometry 2 The 3-dim spherical space forms problem: 3 The 3-dim hyperbolic space forms problem: 4 The 3-dim nilpotent space forms problem: Dekimpe, Igodt, Kim, Lee, Affine structures for closed 3-dimensional manifolds with Nil-geometry, Quart. J. Math. Oxford Ser. (2), 46 (1995),

4 Solvable Space forms Problem Our goal is to do the 3-dim solvable space forms problems, i.e., to classify all the closed 3-manifolds with Sol-geometry up to affine diffeomorphism. These manifolds are infra-solvmanifolds. Their fundamental groups Π are called SB-groups and determine the manifolds completely. Thus we will classify all their fundamental groups. Using elements of group theoretical nature we will be able to write down faithful representations in the affine group Aff(R 3 ) for these groups. In other words, we show, by construction, how the corresponding manifolds can be seen as affinely flat manifolds.

5 Lie group Sol Sol = R 2 σ R where t R acts on R 2 via the maps [ ] e t 0 σ(t) = 0 e t. Sol can be imbedded into Aff(R 3 ) as e t 0 0 x 0 e t 0 y t

6 SB-groups Aff(Sol) = Sol Aut(Sol), the group of affine automorphisms K, a maximal compact subgroup of Aut(Sol) A discrete cocompact subgroup of Sol K Aff(Sol) is called an SC-group, and a torsion-free SC-group is called an SB-group. Examples are crystallographic groups and Bieberbach groups. Aff(R n ) = R n Aut(R n ) = R n GL(n, R) K = O(n) So, R n O(n) = E(R n ) = Isom(R n ).

7 Infra-solvmanifolds For an SB-group E Sol K, the closed manifold E\Sol is called a 3-dimensional infra-solvmanifold. Examples are flat manifolds E\R 3 and infra-nilmanifolds (or almost flat manifolds) E\Nil where E R 3 O(3) and E Nil O(2) are torsion-free discrete cocompact subgroups, respectively.

8 Sol-geometry A closed 3-dimensional manifold M has a Sol-geometry if there is a subgroup Π of Isom(Sol) so that Π acts freely and properly discontinuously with compact quotient M = Π\Sol. CLAIM is that {Manifolds with a Sol-geometry} {Infra-solvmanifolds}, i.e., Isom(Sol) = Sol K where K is a maximal compact subgroup of Aut(Sol).

9 Aut(Sol) Aut(Sol) = Aut 0 (Sol) Z 2 where α 0 γ Aut 0 (Sol) = 0 β δ αβ 0, Z 2 = Thus a maximal compact subgroup of Aut(Sol) is the dihedral group D(4) of order 8 generated by X = = 1 0 0, Y =

10 Isom(Sol) There are two non-equivariant left invariant Riemannian metrics on Sol, and for those metrics the full isometry groups are isomorphic to Sol (Z 2 ) 2 and Sol D(4). K. Y. Ha and J. B. Lee, Left invariant metrics and curvatures on simply connected three-dimensional Lie groups, Math. Nachr., 282 (2009), K. Y. Ha and J. B. Lee, The isometry groups of simply connected 3-diemnsional unimodular Lie groups, submitted for publication. Let K be the compact subgroup (Z 2 ) 2 or D(4) of Aff(Sol). Thus we may assume that E Sol K = Isom(Sol), and since (Z 2 ) 2 D(4), we shall assume in what follows that K = D(4).

11 Associated to Isom(Sol), there is an exact commutative diagram 1 1 R 2 = R 2 1 Sol Isom(Sol) K 1 = 1 R R K K 1 1 1

12 Imbedding Aff(Sol) into Aff(R 3 ) GL(4, R) There is an imbedding λ : Aff(Sol) = Sol Aut(Sol) Aff(R 3 ) GL(4, R): e t 0 0 x αe 0 e t 0 y α 0 µ t 0 0 µe t + x t, 0 β ν 0 βe t 0 νe t + y t e t 0 0 x 0 e 0 e t 0 y t 0 x t, e t 0 0 y t

13 Structure of SC-groups modeled on Sol Theorem (THEOREM A (W. THURSTON)) A group Π is a torsion-free, discrete, cocompact subgroup of Isom(Sol) if and only if Π is torsion-free and contains a lattice Γ Sol of finite index whose centralizer is trivial. Theorem implies that a 3-dimensional closed infra-solvmanifold is finitely covered by a special solvmanifold Γ\Sol. In particular, the 3-dimensional closed infra-solvmanifolds are aspherical.

14 Theorem (THEOREM B (DEKIMPE-K. B. LEE-RAYMOND)) Let G be a connected, simply connected solvable Lie group of type (E) and let C be a compact subgroup of Aut(G). If G has the strong lattice property and if Π is a discrete cocompact subgroup of G C, then Γ = Π G is a lattice of G, and Γ has finite index in Π. type (E) if exp : G G is surjective Corollary If Π is an SC-group modeled on Sol, then Γ = Π Sol is a lattice of Sol, and Γ has finite index in Π. The finite group Φ = Π/Γ is called the holonomy group of Π.

15 Theorem (THEOREM C (K. B. LEE)) Let G be a connected, simply connected solvable Lie group of type (R), and C be a compact subgroup of Aut(G). Let Π, Π G C be discrete cocompact subgroups, which are finite extensions of lattices of G. Then every isomorphism θ : Π Π is a conjugation by an element of G Aut(G). This theorem has been generalized even further to homomorphisms.

16 A covering M M is called essential if no element of the deck transformation group Φ is homotopic to the identity. The map sending a free homotopy class into Out(π 1 (M)) defines a natural homomorphism ρ : Φ Out(π 1 (M)). The covering is essential if and only if ρ is injective. Theorem (THEOREM D (K. B. LEE)) Let G be a connected, simply connected solvable Lie group of type (R), and Γ be a lattice of G. Then there are only finitely many infra-solvmanifolds which are essentially covered by the special solvmanifold Γ\G.

17 Application to Sol Let M = Π\Sol be an infra-solvmanifold which is essentially covered by M = Γ\Sol. Then Γ Π Aff(Sol) and the finite deck transformation group Φ = Π/Γ injects into Out(Γ). For a fixed abstract kernel Φ Out(Γ), Theorem D states that there are only finitely many isomorphism classes of extensions of Γ by Φ, realizing the abstract kernel. Furthermore, if Π = Π, then by Theorem C, they are conjugate to each other by an element of Aff(Sol). This means that the infra-solvmanifolds Π\Sol and Π \Sol are affinely diffeomorphic. Consequently, up to affine diffeomorphism, there are only finitely many infra-solvmanifolds essentially covered by M.

18 Let Π Isom(Sol) = Sol K, an SC-group Γ = Π Sol, a lattice of Sol by Theorems A and B Φ = Π/Γ Since 1 [Sol, Sol] = R 2 Sol Sol/[Sol, Sol] = R 1, taking intersection with Γ, we get Γ [Sol, Sol] = Z 2, Γ/Γ [Sol, Sol] = Z. Let Q = Π/Z 2. Then the diagram before induces the following commutative diagram:

19 1 1 Z 2 Z 2 = 1 Γ Π Φ 1 = 1 Z Q j 1 1 π Φ 1 The exact sequences 1 Z Q Φ 1 in the bottom row and 1 Z 2 Π Q 1 in the middle column will play a significant role in our discussion.

20 Procedure The previous diagram gives rise to homomorphisms φ : Q Aut(Z 2 ) and ψ : Φ Aut(Z), both induced from conjugation by elements of Q. STEP 1 We study the lattices Γ of Sol STEP 2 For each Φ K, we determine all the possible homomorphisms ψ : Φ Aut(Z), and then all the possible extensions Q of Z by Φ with abstract kernel ψ. STEP 3 We classify the (torsion-free) extensions 1 Z 2 Π Q 1 with abstract kernel φ : Q Aut(Z 2 ), which can be imbedded into Aff(R 3 ). STEP 4 We check π has Γ as a finite index subgroup with the trivial centralizer and Π has abstract kernel ρ : Φ Out(Γ) which is an inclusion.

21 Lattices of Sol The following are mainly from J. B. Lee and X. Zho, Nielsen type numbers and homotopy minimal periods for maps on the 3-solvmanifolds, Algebr. Geom. Topol., 8 (2008), Let Γ be a lattice (i.e., a discrete cocompact subgroup) of Sol = R 2 σ R. Then the following diagram of short exact sequences is commutative 1 R 2 Sol R 1 1 Z 2 Γ Z 1

22 Lattices of Sol We may assume that the rightmost injection is the inclusion Z R. Choose a generator t 0 R of Z. Then Z 2 is a σ(t 0 )-invariant lattice of R 2, namely, σ(t 0 ) can be regarded as an automorphism on Z 2. Choose a basis {x 1, x 2 } of Z 2. Then we must have that σ(t 0 )(x i ) = l 1i x 1 + l 2i x 2, (i = 1, 2) for some integers l ij. Let P be the matrix with columns x 1 and x 2 and let [ ] l11 l A = 12. l 21 l 22 Then [ ] PAP 1 e t 0 0 = σ(t 0 ) = 0 e t. 0

23 Remark on A Notice that A SL(2, Z) with trace e t 0 + e t 0 = l 11 + l 22 > 2. This implies that A is a hyperbolic matrix; it has different real eigenvalues: one is greater than 1 and the other is less than 1. Furthermore, neither l 12 nor l 21 vanishes. We denote the lattice Γ by Γ A. Introducing the following notations A(a) = a l 11b l 21 and A(b) = a l 12b l 22, we have Γ A = a, b, t [a, b] = 1, tat 1 = A(a), tbt 1 = A(b).

24 Lattices of Sol Lemma There is a one-to-one correspondence between the set of lattices of Sol modulo isomorphism and the subset of SL(2, Z) with trace > 2 modulo weak conjugacy. Definition An integer matrix B is said to be weakly conjugate to an integer matrix A, written B w A, if B is conjugate to A or A 1 by an integer matrix of determinant ±1.

25 Lattices of Sol Theorem Every hyperbolic element of SL(2, Z) with trace > 2 is weakly conjugate to exactly one element of the following type ( 1) m 1+ +m k (xy) m 1 (xy 1 ) n1 (xy) m k (xy 1 ) n k where m 1, n 1,, m k, n k 1, up to the cyclic permutation rule and up to the interchange rule.

26 Recall 1 1 Z 2 Z 2 = 1 Γ A Π Φ 1 = 1 Z Q j 1 1 π Φ 1 This diagram gives rise to homomorphisms φ : Q Aut(Z 2 ) and ψ : Φ Aut(Z), both induced from conjugation by elements of Q and Φ.

27 Write D(4) = x, y x 4 = y 2 = 1, yxy 1 = x 1. The homomorphism ψ : D(4) Aut(Z) is given by ψ(x) = 1 and ψ(y) = 1. For all nontrivial subgroups Φ of D(4), we will classify the extensions 0 Z Q Φ 1 having ψ = ψ Φ as abstract kernel. For this purpose, we compute Hψ 2 (Φ, Z), and then we simply write out all the possible (inequivalent) presentations for Q corresponding to the elements of Hψ 2 (Φ, Z).

28 Let Φ be a subgroup of D(4) with generators α 1 (and α 2 ). For the simplicity of notation only, we assume Φ has two generators α 1, α 2. For each element α of Φ we fix a unique word u(α) = α i1 α i2 α ir which represents it. So, Φ = α 1, α 2 w i (α 1, α 2 ) = 1 (1 i p).

29 Then, an extension Q of Z by Φ compatible with ψ can be presented as Q = t, α 1, α 2 w i (α 1, α 2 ) = t l i (1 i p), (1) α i tα 1 i = ψ(α i )(t) (i = 1, 2) and the elements q of Q can be written uniquely as words q = t q u(α) (q Z, α Φ). This is completely determined by the set of integers l i. Without confusion, we will abuse the symbol α i as elements of both Q and Φ when α i Q is mapped to α i Φ under the natural quotient map Q Φ.

30 With the help of Hψ 2 (Φ, Z), we can easily find out the possible (inequivalent) extensions Q of Z by Φ having ψ as abstract kernel. There are 12 non-isomorphic Q s!

31 Presentation of E We are given groups Q with presentation of the form (1). In view of computing H 2 φ (Q; Z2 ) in practice, let us consider an extension 0 Z 2 E Q 1 compatible with φ and ψ. Then E has a presentation of the form E = a 1, a 2, t, α 1, α 2 [a 1, a 2 ] = 1 ta i t 1 = A(a i ) (i = 1, 2) α i a j α 1 i = φ(α i )(a j ) (i, j = 1, 2) w i (α 1, α 2 ) = a k 1i 1 ak 2i 2 tl i (1 i p) α i tα 1 i = a k 1i 1 ak 2i 2 ψ(α i)(t) (i = 1, 2) (2).

32 Possible φ Let γ Q be a generator. Write φ(γ) = M Aut(Z 2 ). Then one of the following occurs: CASE 1. γ 2 = 1, γtγ 1 = t and ψ(γ) = 1 CASE 2. γ 2 = t, γtγ 1 = t and ψ(γ) = 1 CASE 3. γ 2 = 1, γtγ 1 = t 1 and ψ(γ) = 1

33 Case 2 CASE 2. γ 2 = t, γtγ 1 = t and ψ(γ) = 1: Then as M 2 = A, M is a square root of A, and MAM 1 = A. Recalling [ ] e t e t = P 1 AP = P 1 M 2 P, 0 we have that M is equal to [ ±P P 1 APP 1 = ± l l11 +l l11 l 21 +l l11 l 12 +l l l11 +l ] if det(m) = 1 ±P [ ] P 1 APP 1 = ± [ l 11 1 l11 +l 22 2 l11 l 21 +l 22 2 l11 l 12 +l 22 2 l 22 1 l11 +l 22 2 ] if det(m) = This rules out some Qs when entries become irrationals.

34 Computational Consistency Given Q, this E is completely determined by the set K of integer vectors k i = [ k 1i k 2i ] t and k i = [ k 1i k 2i] t. But we can not choose these integer vectors completely freely. We refer to E(K ) as the group E determined by K. A set K of integer vectors for which there exists a group E(K ) as an extension of Z 2 by Q is said to be computationally consistent.

35 Special 2-cocycles The elements of E(K ) can be written uniquely as words a p 1 1 ap 2 2 tq u(α) (p 1, p 2, q Z). Take a section s : Q E(K ), q = t q u(α) a1 0a0 2 tq u(α), which we will refer to as the standard section. Definition The cocycle f K : Q Q Z 2 determined by the standard section is called a special cocycle. The set of special cocycles {f K K computationally consistent} will be denoted by SZ 2 φ (Q, Z2 ).

36 Lemma (1) SZ 2 φ (Q, Z2 ) is a subgroup of Z 2 φ (Q, Z2 ). Moreover, if K 1 and K 2 are computationally consistent then f K1 +K 2 = f K1 + f K2. (2) A special cocycle f K is a coboundary if and only if K allows an integer solution to a well determined finite set of matrix equations.

37 Example Given a lattice Γ A of Sol, the short exact sequence is fixed 1 Z 2 Γ A Z 1

38 Among Φ D(4) Aut(Sol) where D(4) = x, y x 4 = y 2 = 1, yxy 1 = x 1. Now we want to complete the bottom exact sequence: 1 Z 2 Γ A 1 Z Q Φ 1 1

39 The bottom exact sequence will induce a homomorphism (abstract kernel) ψ : Φ Aut(Z) For example, we only consider Φ = Z 2 = β, ψ(β) = 1 Then, as Φ D(4), Φ = y, x 2 y, x 2. Because H 2 ψ (Φ, Z) = Z 2, we have Q 1 = t, β β 2 = 1, βtβ 1 = ψ(β)(t) = t = Z Z 2 Q 2 = t, β β 2 = t, βtβ 1 = t = Z

40 Next, for explanation we only consider Q 1. Now we want to complete a commutative diagram of exact λ sequences so that E imbeds into Aff(Sol) Aff(R 3 ): 1 1 Z 2 = Z 2 1 Γ A E Φ 1 = 1 Z Q 1 Φ 1 1 1

41 The extensions E of Z 2 by Q 1 have presentations of the form E(k, k ) = where φ(β) = P a 1, a 2, t, β [a 1, a 2 ] = 1 ta i t 1 = A(a i ) (i = 1, 2) βa i β 1 = φ(β)(a i ) (i = 1, 2), β 2 = a k 1 1 ak 2 2 βtβ 1 = a k 1 1 ak 2 2 ψ(β)(t) [ ] 1 0 P = [ ] The computational consistency condition: k = 0 The coboundary conditions: k = 0, k = 2c(t) + (I A)c(β) for some 1-cochain c : Q 1 Z 2.

42 Consequently, Hφ 2 (Q 1; Z 2 ) = {k Z 2 }/{ 2c(t) + (I A)c(β)} 0 = Z 2 Z 2 Z 2. These information will give us all the inequivalent, at most 4, extensions E(0, k ). In fact, only E(0, 0) can be imbedded into Aff(R 3 ) by taking λ(β) =

43 Conclusion Theorem There are only 7 kinds of SC-groups. Theorem There are only 3 kinds of distinct closed 3-dimensional manifolds M with Sol-geometry. The fundamental group of M is one of the following forms: (1) Γ A = a 1, a 2, t [a 1, a 2 ] = 1, ta i t 1 = A(a i ). (2) Π + A = a 1, a 2, s [a 1, a 2 ] = 1, sa i s 1 = B(a i ) where B is a square root of A with det(b) = 1. (2) Π A = a 1, a 2, s [a 1, a 2 ] = 1, sa i s 1 = B(a i ) where B is a square root of A with det(b) = 1.

44 Many many thanks!