Affine structures for closed 3-dimensional manifolds with Sol-geom
|
|
- Sophie Potter
- 5 years ago
- Views:
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!
AVERAGING FORMULA FOR NIELSEN NUMBERS
Trends in Mathematics - New Series Information Center for Mathematical Sciences Volume 11, Number 1, 2009, pages 71 82 AVERAGING FORMULA FOR NIELSEN NUMBERS JONG BUM LEE AND KYUNG BAI LEE Abstract. We
More informationFormulas for the Reidemeister, Lefschetz and Nielsen Coincidenc. Infra-nilmanifolds
Formulas for the Reidemeister, Lefschetz and Nielsen Coincidence Number of Maps between Infra-nilmanifolds Jong Bum Lee 1 (joint work with Ku Yong Ha and Pieter Penninckx) 1 Sogang University, Seoul, KOREA
More informationALL FOUR-DIMENSIONAL INFRA-SOLVMANIFOLDS ARE BOUNDARIES
ALL FOUR-DIMENSIONAL INFRA-SOLVMANIFOLDS ARE BOUNDARIES SCOTT VAN THUONG Abstract. Infra-solvmanifolds are a certain class of aspherical manifolds which generalize both flat manifolds and almost flat manifolds
More informationNIELSEN THEORY ON NILMANIFOLDS OF THE STANDARD FILIFORM LIE GROUP
NIELSEN THEORY ON NILMANIFOLDS OF THE STANDARD FILIFORM LIE GROUP JONG BUM LEE AND WON SOK YOO Abstract Let M be a nilmanifold modeled on the standard filiform Lie group H m+1 and let f : M M be a self-map
More informationAlmost-Bieberbach groups with prime order holonomy
F U N D A M E N T A MATHEMATICAE 151 (1996) Almost-Bieberbach groups with prime order holonomy by Karel D e k i m p e* and Wim M a l f a i t (Kortrijk) Abstract. The main issue of this paper is an attempt
More informationTitleON THE S^1-FIBRED NILBOTT TOWER. Citation Osaka Journal of Mathematics. 51(1)
TitleON THE S^-FIBRED NILBOTT TOWER Author(s) Nakayama, Mayumi Citation Osaka Journal of Mathematics. 5() Issue 204-0 Date Text Version publisher URL http://hdl.handle.net/094/2987 DOI Rights Osaka University
More informationMAXIMALLY NON INTEGRABLE PLANE FIELDS ON THURSTON GEOMETRIES
MAXIMALLY NON INTEGRABLE PLANE FIELDS ON THURSTON GEOMETRIES T. VOGEL Abstract. We study Thurston geometries (X, G) with contact structures and Engel structures which are compatible with the action of
More informationarxiv: v3 [math.gt] 17 Oct 2010
FORMALITY AND HARD LEFSCHETZ PROPERTIES OF ASPHERICAL MANIFOLDS arxiv:0910.1175v3 [math.gt] 17 Oct 2010 HISASHI KASUYA Abstract. For a virtually polycyclic group Γ, we consider an aspherical manifold M
More informationAn introduction to arithmetic groups. Lizhen Ji CMS, Zhejiang University Hangzhou , China & Dept of Math, Univ of Michigan Ann Arbor, MI 48109
An introduction to arithmetic groups Lizhen Ji CMS, Zhejiang University Hangzhou 310027, China & Dept of Math, Univ of Michigan Ann Arbor, MI 48109 June 27, 2006 Plan. 1. Examples of arithmetic groups
More informationHidden symmetries and arithmetic manifolds
Hidden symmetries and arithmetic manifolds Benson Farb and Shmuel Weinberger Dedicated to the memory of Robert Brooks May 9, 2004 1 Introduction Let M be a closed, locally symmetric Riemannian manifold
More informationMathematische Annalen
Math. Ann. 317, 195 237 (2000) c Springer-Verlag 2000 Mathematische Annalen Rigidity of group actions on solvable Lie groups Burkhard Wilking Received: 27 August 1999 Abstract. We establish analogs of
More informationOn the equivalence of several definitions of compact infra-solvmanifolds
114 Proc. Japan Acad., 89, Ser.A(2013) [Vol.89(A), On the equivalence of several definitions of compact infra-solvmanifolds By Shintarô KUROKI, and Li YU (Communicated by Kenji FUKAYA, M.J.A., Oct. 15,
More informationSEIFERT FIBERINGS AND COLLAPSING OF INFRASOLV SPACES
SEIFERT FIBERINGS AND COLLAPSING OF INFRASOLV SPACES October 31, 2012 OLIVER BAUES AND WILDERICH TUSCHMANN Abstract. We relate collapsing of Riemannian orbifolds and the rigidity theory of Seifert fiberings.
More informationARITHMETICITY OF TOTALLY GEODESIC LIE FOLIATIONS WITH LOCALLY SYMMETRIC LEAVES
ASIAN J. MATH. c 2008 International Press Vol. 12, No. 3, pp. 289 298, September 2008 002 ARITHMETICITY OF TOTALLY GEODESIC LIE FOLIATIONS WITH LOCALLY SYMMETRIC LEAVES RAUL QUIROGA-BARRANCO Abstract.
More informationNilBott Tower of Aspherical Manifolds and Torus Actions
NilBott Tower of Aspherical Manifolds and Torus Actions Tokyo Metropolitan University November 29, 2011 (Tokyo NilBottMetropolitan Tower of Aspherical University) Manifolds and Torus ActionsNovember 29,
More informationALMOST FLAT MANIFOLDS WITH CYCLIC HOLONOMY ARE BOUNDARIES
ALMOST FLAT MANIFOLDS WITH CYCLIC HOLONOMY ARE BOUNDARIES JAMES F. DAVIS AND FUQUAN FANG Abstract. 1. Introduction A closed manifold M is said to be almost flat if there is a sequence of metrics g i on
More informationz, w = z 1 w 1 + z 2 w 2 z, w 2 z 2 w 2. d([z], [w]) = 2 φ : P(C 2 ) \ [1 : 0] C ; [z 1 : z 2 ] z 1 z 2 ψ : P(C 2 ) \ [0 : 1] C ; [z 1 : z 2 ] z 2 z 1
3 3 THE RIEMANN SPHERE 31 Models for the Riemann Sphere One dimensional projective complex space P(C ) is the set of all one-dimensional subspaces of C If z = (z 1, z ) C \ 0 then we will denote by [z]
More informationFlat complex connections with torsion of type (1, 1)
Flat complex connections with torsion of type (1, 1) Adrián Andrada Universidad Nacional de Córdoba, Argentina CIEM - CONICET Geometric Structures in Mathematical Physics Golden Sands, 20th September 2011
More informationEMBEDDING OF THE TEICHMÜLLER SPACE INTO THE GOLDMAN SPACE. Hong Chan Kim. 1. Introduction
J. Korean Math. Soc. 43 006, No. 6, pp. 131 15 EMBEDDING OF THE TEICHMÜLLER SPACE INTO THE GOLDMAN SPACE Hong Chan Kim Abstract. In this paper we shall explicitly calculate the formula of the algebraic
More informationSMALL COVER, INFRA-SOLVMANIFOLD AND CURVATURE
SMALL COVER, INFRA-SOLVMANIFOLD AND CURVATURE SHINTARÔ KUROKI, MIKIYA MASUDA, AND LI YU* Abstract. It is shown that a small cover (resp. real moment-angle manifold) over a simple polytope is an infra-solvmanifold
More informationThe Geometrization Theorem
The Geometrization Theorem Matthew D. Brown Wednesday, December 19, 2012 In this paper, we discuss the Geometrization Theorem, formerly Thurston s Geometrization Conjecture, which is essentially the statement
More informationDifference Cochains and Reidemeister Traces
Chin. Ann. Math. Ser. B 38(6), 2017, 1365 1372 DOI: 10.1007/s11401-017-1044-2 Chinese Annals of Mathematics, Series B c The Editorial Office of CAM and Springer-Verlag Berlin Heidelberg 2017 Difference
More informationFLAT MANIFOLDS WITH HOLONOMY GROUP K 2 DIAGONAL TYPE
Ga sior, A. and Szczepański, A. Osaka J. Math. 51 (214), 115 125 FLAT MANIFOLDS WITH HOLONOMY GROUP K 2 DIAGONAL TYPE OF A. GA SIOR and A. SZCZEPAŃSKI (Received October 15, 212, revised March 5, 213) Abstract
More informationInvariants of knots and 3-manifolds: Survey on 3-manifolds
Invariants of knots and 3-manifolds: Survey on 3-manifolds Wolfgang Lück Bonn Germany email wolfgang.lueck@him.uni-bonn.de http://131.220.77.52/lueck/ Bonn, 10. & 12. April 2018 Wolfgang Lück (MI, Bonn)
More informationRIMS-1897 On multiframings of 3-manifolds By Tatsuro SHIMIZU December 2018 RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES KYOTO UNIVERSITY, Kyoto, Japan
RIMS-1897 On multiframings of 3-manifolds By Tatsuro SHIMIZU December 2018 RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES KYOTO UNIVERSITY, Kyoto, Japan On multiframings of 3 manifolds Tatsuro Shimizu 1
More informationMATRIX LIE GROUPS AND LIE GROUPS
MATRIX LIE GROUPS AND LIE GROUPS Steven Sy December 7, 2005 I MATRIX LIE GROUPS Definition: A matrix Lie group is a closed subgroup of Thus if is any sequence of matrices in, and for some, then either
More informationMAT534 Fall 2013 Practice Midterm I The actual midterm will consist of five problems.
MAT534 Fall 2013 Practice Midterm I The actual midterm will consist of five problems. Problem 1 Find all homomorphisms a) Z 6 Z 6 ; b) Z 6 Z 18 ; c) Z 18 Z 6 ; d) Z 12 Z 15 ; e) Z 6 Z 25 Proof. a)ψ(1)
More informationAN ALMOST KÄHLER STRUCTURE ON THE DEFORMATION SPACE OF CONVEX REAL PROJECTIVE STRUCTURES
Trends in Mathematics Information Center for Mathematical ciences Volume 5, Number 1, June 2002, Pages 23 29 AN ALMOT KÄHLER TRUCTURE ON THE DEFORMATION PACE OF CONVEX REAL PROJECTIVE TRUCTURE HONG CHAN
More informationRepresentations and Linear Actions
Representations and Linear Actions Definition 0.1. Let G be an S-group. A representation of G is a morphism of S-groups φ G GL(n, S) for some n. We say φ is faithful if it is a monomorphism (in the category
More informationCorrections to Introduction to Topological Manifolds (First edition) by John M. Lee December 7, 2015
Corrections to Introduction to Topological Manifolds (First edition) by John M. Lee December 7, 2015 Changes or additions made in the past twelve months are dated. Page 29, statement of Lemma 2.11: The
More informationSurface Groups are Frequently Faithful
Surface Groups are Frequently Faithful Jason DeBlois and Richard P. Kent IV October 22, 2004 Abstract We show the set of faithful representations of a closed orientable surface group is dense in both irreducible
More informationL(C G (x) 0 ) c g (x). Proof. Recall C G (x) = {g G xgx 1 = g} and c g (x) = {X g Ad xx = X}. In general, it is obvious that
ALGEBRAIC GROUPS 61 5. Root systems and semisimple Lie algebras 5.1. Characteristic 0 theory. Assume in this subsection that chark = 0. Let me recall a couple of definitions made earlier: G is called reductive
More informationThe local geometry of compact homogeneous Lorentz spaces
The local geometry of compact homogeneous Lorentz spaces Felix Günther Abstract In 1995, S. Adams and G. Stuck as well as A. Zeghib independently provided a classification of non-compact Lie groups which
More informationX G X by the rule x x g
18. Maps between Riemann surfaces: II Note that there is one further way we can reverse all of this. Suppose that X instead of Y is a Riemann surface. Can we put a Riemann surface structure on Y such that
More informationMostow Rigidity. W. Dison June 17, (a) semi-simple Lie groups with trivial centre and no compact factors and
Mostow Rigidity W. Dison June 17, 2005 0 Introduction Lie Groups and Symmetric Spaces We will be concerned with (a) semi-simple Lie groups with trivial centre and no compact factors and (b) simply connected,
More informationTHE FUNDAMENTAL GROUP OF A COMPACT FLAT LORENTZ SPACE FORM IS VIRTUALLY POLYCYCLIC
J. DIFFERENTIAL GEOMETRY 19 (1984) 233-240 THE FUNDAMENTAL GROUP OF A COMPACT FLAT LORENTZ SPACE FORM IS VIRTUALLY POLYCYCLIC WILLIAM M. GOLDMAN & YOSHINOBU KAMISHIMA A flat Lorentz space form is a geodesically
More informationThe Automorphisms of a Lie algebra
Applied Mathematical Sciences Vol. 9 25 no. 3 2-27 HIKARI Ltd www.m-hikari.com http://dx.doi.org/.2988/ams.25.4895 The Automorphisms of a Lie algebra WonSok Yoo Department of Applied Mathematics Kumoh
More informationTrace fields of knots
JT Lyczak, February 2016 Trace fields of knots These are the knotes from the seminar on knot theory in Leiden in the spring of 2016 The website and all the knotes for this seminar can be found at http://pubmathleidenunivnl/
More informationHOMOLOGY AND COHOMOLOGY. 1. Introduction
HOMOLOGY AND COHOMOLOGY ELLEARD FELIX WEBSTER HEFFERN 1. Introduction We have been introduced to the idea of homology, which derives from a chain complex of singular or simplicial chain groups together
More informationarxiv:math/ v1 [math.rt] 1 Jul 1996
SUPERRIGID SUBGROUPS OF SOLVABLE LIE GROUPS DAVE WITTE arxiv:math/9607221v1 [math.rt] 1 Jul 1996 Abstract. Let Γ be a discrete subgroup of a simply connected, solvable Lie group G, such that Ad G Γ has
More informationNOTES ON FIBER BUNDLES
NOTES ON FIBER BUNDLES DANNY CALEGARI Abstract. These are notes on fiber bundles and principal bundles, especially over CW complexes and spaces homotopy equivalent to them. They are meant to supplement
More informationarxiv: v1 [math.dg] 4 Sep 2008
arxiv:0809.0824v1 [math.dg] 4 Sep 2008 Prehomogeneous Affine Representations and Flat Pseudo-Riemannian Manifolds Oliver Baues Institut für Algebra und Geometrie Universität Karlsruhe D-76128 Karlsruhe
More informationNotation. For any Lie group G, we set G 0 to be the connected component of the identity.
Notation. For any Lie group G, we set G 0 to be the connected component of the identity. Problem 1 Prove that GL(n, R) is homotopic to O(n, R). (Hint: Gram-Schmidt Orthogonalization.) Here is a sequence
More informationAffine Geometry and Hyperbolic Geometry
Affine Geometry and Hyperbolic Geometry William Goldman University of Maryland Lie Groups: Dynamics, Rigidity, Arithmetic A conference in honor of Gregory Margulis s 60th birthday February 24, 2006 Discrete
More information1 v >, which will be G-invariant by construction.
1. Riemannian symmetric spaces Definition 1.1. A (globally, Riemannian) symmetric space is a Riemannian manifold (X, g) such that for all x X, there exists an isometry s x Iso(X, g) such that s x (x) =
More informationDeformation of Properly Discontinuous Actions of Z k on R k+1
Deformation of Properly Discontinuous Actions of Z k on R k+1 Toshiyuki KOBAYASHI Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502, Japan E-mail address: toshi@kurims.kyoto-u.ac.jp
More informationAll nil 3-manifolds are cusps of complex hyperbolic 2-orbifolds
All nil 3-manifolds are cusps of complex hyperbolic 2-orbifolds D. B. McReynolds September 1, 2003 Abstract In this paper, we prove that every closed nil 3-manifold is diffeomorphic to a cusp cross-section
More informationSpherical three-dimensional orbifolds
Spherical three-dimensional orbifolds Andrea Seppi joint work with Mattia Mecchia Pisa, 16th May 2013 Outline What is an orbifold? What is a spherical orientable orbifold? What is a fibered orbifold? An
More informationTHE CREMONA GROUP: LECTURE 1
THE CREMONA GROUP: LECTURE 1 Birational maps of P n. A birational map from P n to P n is specified by an (n + 1)-tuple (f 0,..., f n ) of homogeneous polynomials of the same degree, which can be assumed
More informationPart II. Algebraic Topology. Year
Part II Year 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2017 Paper 3, Section II 18I The n-torus is the product of n circles: 5 T n = } S 1. {{.. S } 1. n times For all n 1 and 0
More informationOn Shalom Tao s Non-Quantitative Proof of Gromov s Polynomial Growth Theorem
On Shalom Tao s Non-Quantitative Proof of Gromov s Polynomial Growth Theorem Carlos A. De la Cruz Mengual Geometric Group Theory Seminar, HS 2013, ETH Zürich 13.11.2013 1 Towards the statement of Gromov
More informationTHE EULER CHARACTERISTIC OF A LIE GROUP
THE EULER CHARACTERISTIC OF A LIE GROUP JAY TAYLOR 1 Examples of Lie Groups The following is adapted from [2] We begin with the basic definition and some core examples Definition A Lie group is a smooth
More informationCollisions at infinity in hyperbolic manifolds
Under consideration for publication in Math. Proc. Camb. Phil. Soc. 1 Collisions at infinity in hyperbolic manifolds By D. B. MCREYNOLDS Department of Mathematics, Purdue University, Lafayette, IN 47907,
More informationOn the Dimension of the Stability Group for a Levi Non-Degenerate Hypersurface
1 On the Dimension of the Stability Group for a Levi Non-Degenerate Hypersurface Vladimir Ezhov and Alexander Isaev We classify locally defined non-spherical real-analytic hypersurfaces in complex space
More informationMath 121 Homework 4: Notes on Selected Problems
Math 121 Homework 4: Notes on Selected Problems 11.2.9. If W is a subspace of the vector space V stable under the linear transformation (i.e., (W ) W ), show that induces linear transformations W on W
More informationCompactifications of Margulis space-times
1/29 Compactifications of Margulis space-times Suhyoung Choi Department of Mathematical Science KAIST, Daejeon, South Korea mathsci.kaist.ac.kr/ schoi email: schoi@math.kaist.ac.kr Geometry of Moduli Spaces
More informationSolutions of exercise sheet 8
D-MATH Algebra I HS 14 Prof. Emmanuel Kowalski Solutions of exercise sheet 8 1. In this exercise, we will give a characterization for solvable groups using commutator subgroups. See last semester s (Algebra
More informationAFFINE ACTIONS ON NILPOTENT LIE GROUPS
AFFINE ACTIONS ON NILPOTENT LIE GROUPS DIETRICH BURDE, KAREL DEKIMPE, AND SANDRA DESCHAMPS Abstract. To any connected and simply connected nilpotent Lie group N, one can associate its group of affine transformations
More informationMath 121 Homework 5: Notes on Selected Problems
Math 121 Homework 5: Notes on Selected Problems 12.1.2. Let M be a module over the integral domain R. (a) Assume that M has rank n and that x 1,..., x n is any maximal set of linearly independent elements
More informationSOLVABLE GROUPS OF EXPONENTIAL GROWTH AND HNN EXTENSIONS. Roger C. Alperin
SOLVABLE GROUPS OF EXPONENTIAL GROWTH AND HNN EXTENSIONS Roger C. Alperin An extraordinary theorem of Gromov, [Gv], characterizes the finitely generated groups of polynomial growth; a group has polynomial
More informationMaster Algèbre géométrie et théorie des nombres Final exam of differential geometry Lecture notes allowed
Université de Bordeaux U.F. Mathématiques et Interactions Master Algèbre géométrie et théorie des nombres Final exam of differential geometry 2018-2019 Lecture notes allowed Exercise 1 We call H (like
More informationRECOGNIZING GEOMETRIC 3 MANIFOLD GROUPS USING THE WORD PROBLEM
RECOGNIZING GEOMETRIC 3 MANIFOLD GROUPS USING THE WORD PROBLEM DANIEL P. GROVES, JASON FOX MANNING, AND HENRY WILTON Abstract. Adyan and Rabin showed that most properties of groups cannot be algorithmically
More informationInfinite generation of non-cocompact lattices on right-angled buildings
Infinite generation of non-cocompact lattices on right-angled buildings ANNE THOMAS KEVIN WORTMAN Let Γ be a non-cocompact lattice on a locally finite regular right-angled building X. We prove that if
More informationREPRESENTING THE AUTOMORPHISM GROUP OF AN ALMOST CRYSTALLOGRAPHIC GROUP
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Xxxx 9XX REPRESENTING THE AUTOMORPHISM GROUP OF AN ALMOST CRYSTALLOGRAPHIC GROUP PAUL IGODT AND WIM MALFAIT (Communicated by Ron Solomon)
More informationAN ASPHERICAL 5-MANIFOLD WITH PERFECT FUNDAMENTAL GROUP
AN ASPHERICAL 5-MANIFOLD WITH PERFECT FUNDAMENTAL GROUP J.A. HILLMAN Abstract. We construct aspherical closed orientable 5-manifolds with perfect fundamental group. This completes part of our study of
More informationIsometries of Riemannian and sub-riemannian structures on 3D Lie groups
Isometries of Riemannian and sub-riemannian structures on 3D Lie groups Rory Biggs Geometry, Graphs and Control (GGC) Research Group Department of Mathematics, Rhodes University, Grahamstown, South Africa
More informationNoetherian property of infinite EI categories
Noetherian property of infinite EI categories Wee Liang Gan and Liping Li Abstract. It is known that finitely generated FI-modules over a field of characteristic 0 are Noetherian. We generalize this result
More informationOn groups of diffeomorphisms of the interval with finitely many fixed points I. Azer Akhmedov
On groups of diffeomorphisms of the interval with finitely many fixed points I Azer Akhmedov Abstract: We strengthen the results of [1], consequently, we improve the claims of [2] obtaining the best possible
More informationLecture 22 - F 4. April 19, The Weyl dimension formula gives the following dimensions of the fundamental representations:
Lecture 22 - F 4 April 19, 2013 1 Review of what we know about F 4 We have two definitions of the Lie algebra f 4 at this point. The old definition is that it is the exceptional Lie algebra with Dynkin
More informationNotes 10: Consequences of Eli Cartan s theorem.
Notes 10: Consequences of Eli Cartan s theorem. Version 0.00 with misprints, The are a few obvious, but important consequences of the theorem of Eli Cartan on the maximal tori. The first one is the observation
More informationCopyright by Samuel Aaron Ballas 2013
Copyright by Samuel Aaron Ballas 2013 The Dissertation Committee for Samuel Aaron Ballas certifies that this is the approved version of the following dissertation: Flexibility and Rigidity of Three-Dimensional
More informationDifferential Rigidity of Anosov Actions of Higher Rank Abelian Groups and Algebraic Lattice Actions
Differential Rigidity of Anosov Actions of Higher Rank Abelian Groups and Algebraic Lattice Actions A. Katok and R. J. Spatzier To Dmitry Viktorovich Anosov on his sixtieth birthday Abstract We show that
More informationLecture 6: Etale Fundamental Group
Lecture 6: Etale Fundamental Group October 5, 2014 1 Review of the topological fundamental group and covering spaces 1.1 Topological fundamental group Suppose X is a path-connected topological space, and
More information5. Crystallography. A v { A GL(n, R), v E n } 0 1
5. Crystallography Let E n denote n-dimensional Euclidean space and let R(n) be the group of rigid motions of E n. We observe that R(n) contains the following elements: (a) all translations. For each vector
More informationERRATA. Abstract Algebra, Third Edition by D. Dummit and R. Foote (most recently revised on March 4, 2009)
ERRATA Abstract Algebra, Third Edition by D. Dummit and R. Foote (most recently revised on March 4, 2009) These are errata for the Third Edition of the book. Errata from previous editions have been fixed
More informationDeformations and rigidity of lattices in solvable Lie groups
1 Deformations and rigidity of lattices in solvable Lie groups joint work with Benjamin Klopsch Oliver Baues Institut für Algebra und Geometrie, Karlsruher Institut für Technologie (KIT), 76128 Karlsruhe,
More information(E.-W. Zink, with A. Silberger)
1 Langlands classification for L-parameters A talk dedicated to Sergei Vladimirovich Vostokov on the occasion of his 70th birthday St.Petersburg im Mai 2015 (E.-W. Zink, with A. Silberger) In the representation
More informationA Primer on Homological Algebra
A Primer on Homological Algebra Henry Y Chan July 12, 213 1 Modules For people who have taken the algebra sequence, you can pretty much skip the first section Before telling you what a module is, you probably
More informationjoint work with A. Fino, M. Macrì, G. Ovando, M. Subils Rosario (Argentina) July 2012 Sergio Console Dipartimento di Matematica Università di Torino
joint work with A. Fino, M. Macrì, G. Ovando, M. Subils Rosario (Argentina) July 2012 Proof of the structures on Dipartimento di Matematica Università di Torino 1 1 2 Lie groups 3 de Rham cohomology of
More informationRank 3 Latin square designs
Rank 3 Latin square designs Alice Devillers Université Libre de Bruxelles Département de Mathématiques - C.P.216 Boulevard du Triomphe B-1050 Brussels, Belgium adevil@ulb.ac.be and J.I. Hall Department
More informationMath 249B. Geometric Bruhat decomposition
Math 249B. Geometric Bruhat decomposition 1. Introduction Let (G, T ) be a split connected reductive group over a field k, and Φ = Φ(G, T ). Fix a positive system of roots Φ Φ, and let B be the unique
More informationMath 210C. A non-closed commutator subgroup
Math 210C. A non-closed commutator subgroup 1. Introduction In Exercise 3(i) of HW7 we saw that every element of SU(2) is a commutator (i.e., has the form xyx 1 y 1 for x, y SU(2)), so the same holds for
More informationCitation Osaka Journal of Mathematics. 43(1)
TitleA note on compact solvmanifolds wit Author(s) Hasegawa, Keizo Citation Osaka Journal of Mathematics. 43(1) Issue 2006-03 Date Text Version publisher URL http://hdl.handle.net/11094/11990 DOI Rights
More informationSome notes on Coxeter groups
Some notes on Coxeter groups Brooks Roberts November 28, 2017 CONTENTS 1 Contents 1 Sources 2 2 Reflections 3 3 The orthogonal group 7 4 Finite subgroups in two dimensions 9 5 Finite subgroups in three
More informationarxiv: v1 [math.dg] 19 Mar 2018
MAXIMAL SYMMETRY AND UNIMODULAR SOLVMANIFOLDS MICHAEL JABLONSKI arxiv:1803.06988v1 [math.dg] 19 Mar 2018 Abstract. Recently, it was shown that Einstein solvmanifolds have maximal symmetry in the sense
More informationBASIC GROUP THEORY : G G G,
BASIC GROUP THEORY 18.904 1. Definitions Definition 1.1. A group (G, ) is a set G with a binary operation : G G G, and a unit e G, possessing the following properties. (1) Unital: for g G, we have g e
More informationMATH 436 Notes: Cyclic groups and Invariant Subgroups.
MATH 436 Notes: Cyclic groups and Invariant Subgroups. Jonathan Pakianathan September 30, 2003 1 Cyclic Groups Now that we have enough basic tools, let us go back and study the structure of cyclic groups.
More informationON NEARLY SEMIFREE CIRCLE ACTIONS
ON NEARLY SEMIFREE CIRCLE ACTIONS DUSA MCDUFF AND SUSAN TOLMAN Abstract. Recall that an effective circle action is semifree if the stabilizer subgroup of each point is connected. We show that if (M, ω)
More informationarxiv:math/ v3 [math.gt] 24 Oct 2005
GENERALIZED HANTZSCHE-WENDT FLAT MANIFOLDS arxiv:math/0208205v3 [math.gt] 24 Oct 2005 J. P. ROSSETTI ( ) AND A. SZCZEPAŃSKI Abstract. We study the family of closed Riemannian n-manifolds with holonomy
More informationBIEBERBACH THEOREMS FOR SOLVABLE LIE GROUPS*
ASIAN J. MATH. 2001 International Press Vol. 5, No. 3, pp. 499-508, September 2001 006 BIEBERBACH THEOREMS FOR SOLVABLE LIE GROUPS* KAREL DEKIMPEt, KYUNG BAI LEE*, AND FRANK RAYMOND^ 1. Introduction. Let
More informationThe Outer Automorphism of S 6
Meena Jagadeesan 1 Karthik Karnik 2 Mentor: Akhil Mathew 1 Phillips Exeter Academy 2 Massachusetts Academy of Math and Science PRIMES Conference, May 2016 What is a Group? A group G is a set of elements
More informationarxiv:math/ v3 [math.gr] 21 Feb 2006
arxiv:math/0407446v3 [math.gr] 21 Feb 2006 Separability of Solvable Subgroups in Linear Groups Roger Alperin and Benson Farb August 11, 2004 Abstract Let Γ be a finitely generated linear group over a field
More informationCONFORMALLY FLAT MANIFOLDS WITH NILPOTENT HOLONOMY AND THE UNIFORMIZATION PROBLEM FOR 3-MANIFOLDS
transactions of the american mathematical Volume 278, Number 2, August 1983 society CONFORMALLY FLAT MANIFOLDS WITH NILPOTENT HOLONOMY AND THE UNIFORMIZATION PROBLEM FOR 3-MANIFOLDS BY WILLIAM M. GOLDMAN
More information3-manifolds and their groups
3-manifolds and their groups Dale Rolfsen University of British Columbia Marseille, September 2010 Dale Rolfsen (2010) 3-manifolds and their groups Marseille, September 2010 1 / 31 3-manifolds and their
More informationCS 468: Computational Topology Group Theory Fall b c b a b a c b a c b c c b a
Q: What s purple and commutes? A: An abelian grape! Anonymous Group Theory Last lecture, we learned about a combinatorial method for characterizing spaces: using simplicial complexes as triangulations
More informationBredon, Introduction to compact transformation groups, Academic Press
1 Introduction Outline Section 3: Topology of 2-orbifolds: Compact group actions Compact group actions Orbit spaces. Tubes and slices. Path-lifting, covering homotopy Locally smooth actions Smooth actions
More informationLeft-invariant Einstein metrics
on Lie groups August 28, 2012 Differential Geometry seminar Department of Mathematics The Ohio State University these notes are posted at http://www.math.ohio-state.edu/ derdzinski.1/beamer/linv.pdf LEFT-INVARIANT
More informationStable complex and Spin c -structures
APPENDIX D Stable complex and Spin c -structures In this book, G-manifolds are often equipped with a stable complex structure or a Spin c structure. Specifically, we use these structures to define quantization.
More informationMath 215B: Solutions 1
Math 15B: Solutions 1 Due Thursday, January 18, 018 (1) Let π : X X be a covering space. Let Φ be a smooth structure on X. Prove that there is a smooth structure Φ on X so that π : ( X, Φ) (X, Φ) is an
More informationAn introduction to orbifolds
An introduction to orbifolds Joan Porti UAB Subdivide and Tile: Triangulating spaces for understanding the world Lorentz Center November 009 An introduction to orbifolds p.1/0 Motivation Γ < Isom(R n )
More information