## complex and quantum tori (2009)

### BibTeX

@MISC{Nikolaev09complexand,

author = {Igor Nikolaev},

title = {complex and quantum tori},

year = {2009}

}

### OpenURL

### Abstract

noncommutative geometry I: classification of

### Citations

2276 | Non-commutative Geometry
- Connes, A
- 1994
(Show Context)
Citation Context ...2g , ϕk), ϕk ∈ SL(2g, Z + ). (1) G parametrizes T(g, n) on the one hand, and represents K0-group of a C ∗ - algebra BG, to the other. Set X = BG can be viewed as a noncommutative space, see A. Connes =-=[1]-=-, Chapter II. G is an ordered abelian group Z 2g . The order is specified by a facet of hyperplanes passing through the origin of Z 2g , see Effros [3]. “Generic” G are totally ordered, i.e. the facet... |

940 |
The Arithmetic of Elliptic Curves
- Silverman
- 1986
(Show Context)
Citation Context ... review C ∗ -algebras, K-theory, dimension groups and complex multiplication of elliptic curves. The reader can find additional information in Effros [3], Rørdam, Larsen & Laustsen [17] and Silverman =-=[18]-=-. 4.1 K-theory of C*-algebras By the C ∗ -algebra one understands a noncommutative Banach algebra with an involution [17]. Namely, a C ∗ -algebra A is an algebra over C with a norm a ↦→ ||a|| and an i... |

389 |
Intersection theory on the moduli space of curves and the matrix Airy function
- Kontsevich
- 1992
(Show Context)
Citation Context .... Hain & Looijenga [6]. Mumford suggested a triangulation of T(g, n) based on quadratic differentials and ribbon graphs, see [6]. Exactly this approach allowed Kontsevich to prove Witten’s conjecture =-=[11]-=-. Hatcher and Thurston introduced a complex of curves C(S) to study action of the mapping class group [9]. Harer calculated homotopy and homology groups of M(g, n) using such complexes, see survey [7]... |

387 | The Irreducibility of the Space of Curves of Given Genus
- Deligne
(Show Context)
Citation Context ...lass group of S. Whereas T(g, n) is homeomorphic to open ball D 6g−6+2n , topology of M(g, n) is largely a mystery. M(g, n) is an irreducible quasi-projective variety as proved by Deligne and Mumford =-=[2]-=-. Since the action of Γ(g, n) is properly discontinuous, M(g, n) is in fact smooth manifold except “corners”. Much of topology of variety M(g, n) is encrypted by “tautological” cohomology classes, cf.... |

194 |
Analytic Theory of Continued Fractions
- Wall
- 1967
(Show Context)
Citation Context ...r orientation preserving automorphisms, e.g. this can be accomplished by fixing det gk+1 = +1. It is well known that every ordered sequence of gi ∈ SL(2, Z) unfolds into a continued fraction, cf Wall =-=[19]-=-. The convergents of this fraction can be regarded as “directional averaging” of the geodesic sequence {γ1, γ2, . . .}. Namely, every matrix (gk) ∈ SL(2, Z) admits a unique decomposition ( ) ( −q ∓s 0... |

100 |
A presentation for the mapping class group of a closed orientable surface, Topology 19
- Hatcher, Thurston
- 1980
(Show Context)
Citation Context ...ribbon graphs, see [6]. Exactly this approach allowed Kontsevich to prove Witten’s conjecture [11]. Hatcher and Thurston introduced a complex of curves C(S) to study action of the mapping class group =-=[9]-=-. Harer calculated homotopy and homology groups of M(g, n) using such complexes, see survey [7]. Homology of M(g, n) stabilizes, i.e. doesn’t depend on g when g >> 0. Such a property is typical for ho... |

45 | A fundamental class of geodesics on any closed surface of genus greater than one - Morse - 1924 |

40 |
Selberg’s trace formula as applied to a compact Riemann surface
- McKean
- 1972
(Show Context)
Citation Context ...t = Area (E) 4πt ∑ L |ω|2 − e 4t , (4) where L ′ is the lattice dual to L and ω ′ ∈ L ′ . This allows to calculate a = Area E and b through unfolding the Poisson formula into the power series (McKean =-=[13]-=-). □ Let Spec S = {l1, l2, . . .} be length spectrum of a Riemann surface S. Let a > 0 be a real number. By aSpec S we understand the length spectrum {al1, al2, . . .}. Similarly, for any m ∈ N we den... |

39 |
An Introduction to K-theory for
- Rørdam, Larsen, et al.
- 2000
(Show Context)
Citation Context ... section we briefly review C ∗ -algebras, K-theory, dimension groups and complex multiplication of elliptic curves. The reader can find additional information in Effros [3], Rørdam, Larsen & Laustsen =-=[17]-=- and Silverman [18]. 4.1 K-theory of C*-algebras By the C ∗ -algebra one understands a noncommutative Banach algebra with an involution [17]. Namely, a C ∗ -algebra A is an algebra over C with a norm ... |

36 |
The cohomology of the moduli space of curves, in: Theory of Moduli, E.Sernesi ed
- Harer
- 1988
(Show Context)
Citation Context ...11]. Hatcher and Thurston introduced a complex of curves C(S) to study action of the mapping class group [9]. Harer calculated homotopy and homology groups of M(g, n) using such complexes, see survey =-=[7]-=-. Homology of M(g, n) stabilizes, i.e. doesn’t depend on g when g >> 0. Such a property is typical for homology of arithmetic groups, what inspired Harvey to conjecture that Γ(g, n) is arithmetic, see... |

30 |
Travaux de Thurston sur les surfaces, Seminaire Orsay", Asterisque
- Fathi, Laudenbach, et al.
- 1979
(Show Context)
Citation Context ... Then “generically” there exists a one-to-one mapping T(g, 0) → G, where G is the space of simple dimension groups of rank 2g. Proof. First let us prove the following lemma due to Fathi and Shub (see =-=[5]-=-, Exposé 10): Lemma 1 Let h : S → S be a pseudo-Anosov homeomorphism of the compact surface of genus g ≥ 1. Then there exists a Markov partition of S based on 2g elements. Proof. Proof of this fact fo... |

29 |
Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse. Inanguraldissertation, Göttingen 1851. In: Bernhard Riemann’s gesammelte mathematische Werke und wissenschaftlicher Nachlass. Hrsg
- RIEMANN
(Show Context)
Citation Context ...elliptic curve, noncommutative torus AMS (MOS) Subj. Class.: 14H52, 46L85, 57M50 Introduction Search for conformal invariants of two-dimensional manifolds has a long history and honorable origin, see =-=[16]-=-. Except for the cases g = 0, 1, 2 no satisfactory set of such invariants is known, despite many efforts and evident progress, see e.g. Eisenbud & Harris [4]. 1Recall that surface S = S(g, n) with g ... |

26 |
Geometric structure of surface mapping class groups. Homological group theory (Proc. Sympos
- Harvey
- 1977
(Show Context)
Citation Context ... Homology of M(g, n) stabilizes, i.e. doesn’t depend on g when g >> 0. Such a property is typical for homology of arithmetic groups, what inspired Harvey to conjecture that Γ(g, n) is arithmetic, see =-=[8]-=- p. 267. Harvey’s conjecture isn’t true in general, as shown by N. V. Ivanov, see [7]. Still Γ(g, n) bears many traits of arithmetic groups, as confirmed by the remarkable formula χ(M(g, n)) = (−1) n−... |

9 |
Progress in the theory of complex algebraic curves
- Eisenbud, Harris
- 1989
(Show Context)
Citation Context ... a long history and honorable origin, see [16]. Except for the cases g = 0, 1, 2 no satisfactory set of such invariants is known, despite many efforts and evident progress, see e.g. Eisenbud & Harris =-=[4]-=-. 1Recall that surface S = S(g, n) with g handles and n holes is topologically unique. It is no longer true that S is conformally unique. Riemann found that there exists T(g, n) ≃ R 6g−6+2n different... |

8 |
Multiplication and noncommutative geometry, arXiv math
- Manin, Real
(Show Context)
Citation Context ...re Morita equivalent whenever θ ′ = aθ+b cθ+d , ( ) a b ∈ SL(2, Z). Comparing Eτ with Tθ suggests an important c d analogy between moduli of complex torus and geometry of quantum tori, see e.g. Manin =-=[12]-=-. Present note is an attempt to show that this is not an isolated fact, but 2indeed a part of general theory. Namely, let Spec S be the length spectrum of S. Length spectrum is a conformal invariant ... |

7 |
Ausgewählte Kapitel der Zahlentheorie
- Klein
- 1907
(Show Context)
Citation Context ...ons of α, until all closed geodesics of E are exhausted. The conclusion of Lemma 7 follows. □ We shall need the following statement regarding geometry of the regular continued fractions, see F. Klein =-=[10]-=-. Lemma 8 (F. Klein) Let 1 ω = µ1 + 1 µ2 + µ3 + . . . (10) be a regular continued fraction. Let us denote the convergents of ω by: p−1 q−1 = 0 1 , p0 q0 = 1 0 , p1 q1 = µ1 pν , . . ., 1 qν = µνpν−1 + ... |

1 |
Moduli of complex curves and noncommutative geometry II: arithmeticity of the mapping class group (in preparation
- Nikolaev
(Show Context)
Citation Context ...d that this is not true generally speaking. In this section we apply our analysis to prove Harvey’s conjecture for at least “generic” Riemann surfaces. (We expect to give a more detailed treatment in =-=[15]-=-.) Theorem 3 The mapping class group Γ(g, 0) is “typically” arithmetic. In other words, to every action of Γ(g, 0) on “typical” Riemann surfaces S ∈ T(g, n), there exists a representation of Γ(g, 0) a... |