## (2009)

### Abstract

Moduli of complex curves and noncommutative geometry: Riemann surfaces and dimension groups

### Citations

346 |
The irreducibility of the space of curves of given genus
- Deligne, Mumford
- 1969
(Show Context)
Citation Context ...ss 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 1by Deligne and Mumford =-=[3]-=-. 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.... |

122 |
Guass measures for transformations on the space of interval exchange maps
- Veech
- 1982
(Show Context)
Citation Context ...on of such groups). Connection between the Riemann surfaces, quadratic differentials and interval exchange transformation has been discussed by many authors, see Bödigheimer [1], Masur [18] and Veech =-=[22]-=-. First we remind the reader how this connection comes in, and next we use it to define dimension groups associated to Riemann surfaces. 1.1 Interval exchange transformations Let m ≥ 2 be a positive i... |

114 |
Interval exchange transformations and measured foliations
- Masur
- 1982
(Show Context)
Citation Context ...or the definition of such groups). Connection between the Riemann surfaces, quadratic differentials and interval exchange transformation has been discussed by many authors, see Bödigheimer [1], Masur =-=[18]-=- and Veech [22]. First we remind the reader how this connection comes in, and next we use it to define dimension groups associated to Riemann surfaces. 1.1 Interval exchange transformations Let m ≥ 2 ... |

102 |
On the classification of inductive limits of sequences of semi-simple finite-dimensional algebras
- Elliott
- 1976
(Show Context)
Citation Context ...nd to a “positive cone” K + 0 ⊂ K0(A) and the unit element 1 ∈ A corresponds to an “order unit” [1] ∈ K0(A). The ordered abelian group (K0, K + 0 , [1]) with an order unit is called a dimension group =-=[6]-=-. 3.2 Dimension groups We use notation Z, Z + , Q and R for integers, positive integers, rationals and reals, respectively and GLn(Z) for the group of n × n matrices with entries in Z and determinant ... |

93 |
A presentation for the mapping class group of a closed orientable surface, Topology 19
- Hatcher, Thurston
- 1980
(Show Context)
Citation Context ...ribbon graphs, see [8]. Exactly this approach allowed Kontsevich to prove Witten’s conjecture [15]. Hatcher and Thurston introduced a complex of curves C(S) to study action of the mapping class group =-=[11]-=-. Harer calculated homotopy and homology groups of M(g, n) using such complexes, see [9]. Homology of M(g, n) stabilizes, i.e. doesn’t depend on g when g >> 0. Such a property is typical for homology ... |

88 |
Interval exchange transformations
- Keane
- 1975
(Show Context)
Citation Context ...ths of intervals in the I.E.T. associated to S. It was proved by Keane that a sufficient condition to I.E.T. to be minimal consists in that numbers λ1, . . . , λm are linearly independent over Q, see =-=[14]-=-. Consider abelian subgroup of the real line: Γ = Zλ1 + . . . + Zλm ⊂ R. (9) It is clear that Γ has rank m and by Keane’s condition is dense in R. Then by the pull-back construction, Γ generates a sim... |

65 | Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Inventiones mathematicæ
- Kontsevich
(Show Context)
Citation Context ...ding to quadratic differentials with fixed singularity data, a stratum Hi. By the index argument, the number of strata is finite. Strata aren’t generally connected, as shown by Kontsevitch and Zorich =-=[16]-=-. If k1, . . . , kp is the singularity data of stratum Hi, then the number m of intervals in the exchange map: m = { 2g + p − 1 if q is Abelian 2g + p − 2 otherwise, (6) see Veech [23]. The last formu... |

38 |
An introduction to K-theory for
- Rørdam, Larsen, et al.
(Show Context)
Citation Context ...Theorem 1 follows. □ 3 Appendix In this section we briefly review C ∗ -algebras, K-theory and dimension groups. The reader can find additional information in Effros [5], and Rørdam, Larsen & Laustsen =-=[21]-=-. 93.1 K-theory of C*-algebras By the C ∗ -algebra one understands a noncommutative Banach algebra with an involution [21]. Namely, a C ∗ -algebra A is an algebra over C with a norm a ↦→ ||a|| and an... |

34 |
The cohomology of the moduli space of curves, in Theory of moduli
- Harer
- 1988
(Show Context)
Citation Context ...ture [15]. Hatcher and Thurston introduced a complex of curves C(S) to study action of the mapping class group [11]. Harer calculated homotopy and homology groups of M(g, n) using such complexes, see =-=[9]-=-. 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... |

32 | Mapping class groups and moduli spaces of curves
- Hain, Looijenga
- 1997
(Show Context)
Citation Context ... 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. Hain & Looijenga =-=[8]-=-. Mumford suggested a triangulation of T(g, n) based on quadratic differentials and ribbon graphs, see [8]. Exactly this approach allowed Kontsevich to prove Witten’s conjecture [15]. Hatcher and Thur... |

30 |
Moduli spaces of quadratic Differentials
- Veech
(Show Context)
Citation Context ...itch and Zorich [16]. If k1, . . . , kp is the singularity data of stratum Hi, then the number m of intervals in the exchange map: m = { 2g + p − 1 if q is Abelian 2g + p − 2 otherwise, (6) see Veech =-=[23]-=-. The last formula allows to evaluate the dimension of strata. The stratum of maximal dimension is given by non-Abelian non-Strebel quadratic differential with “simple” singularities (tripods) whose t... |

26 |
Geometric structure of surface mapping class groups, from: “Homological group theory
- Harvey
- 1979
(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 =-=[10]-=- p. 267. Harvey’s conjecture isn’t true in general, as it was shown by N. V. Ivanov [13]. Still Γ(g, n) bears many traits of arithmetic groups, as confirmed by the remarkable formula χ(M(g, n)) = (−1)... |

25 |
Grundlagen für eine allgemeine Theorie der Funktionen einer veränderlichen komplexen grösse (1851) in collected works
- Riemann
(Show Context)
Citation Context ...rases: Riemann surface, dimension group AMS (MOS) Subj. Class.: 14H52, 46L85 Introduction Search for the conformal invariants of two-dimensional manifolds has a long history and honorable origin, see =-=[20]-=-. 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 [7]. Recall that surface S = S(g, n) with g ha... |

18 |
Algebraic properties of the Teichmüller modular group
- Ivanov
- 1984
(Show Context)
Citation Context ...s typical for homology of arithmetic groups, what inspired Harvey to conjecture that Γ(g, n) is arithmetic, see [10] p. 267. Harvey’s conjecture isn’t true in general, as it was shown by N. V. Ivanov =-=[13]-=-. Still Γ(g, n) bears many traits of arithmetic groups, as confirmed by the remarkable formula χ(M(g, n)) = (−1) n−1 (2g+n−3)! (2g−2)! ζ(1 − 2g) of Harer & Zagier. (Here χ is the orbifold Euler charac... |

9 |
Progress in the theory of complex algebraic curves
- Eisenbud, Harris
- 1989
(Show Context)
Citation Context ... a long history and honorable origin, see [20]. 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 =-=[7]-=-. Recall 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 c... |

7 |
Multiplication and noncommutative geometry, arXiv math
- Manin, Real
(Show Context)
Citation Context ...ori are order-isomorphic whenever θ ′ = aθ+b cθ+d , ( ) a b ∈ SL(2, Z). Comparing Eτ with Gθ suggests an c d important morphism between moduli of complex torus and geometry of quantum tori, see Manin =-=[17]-=-. Present note is an attempt to “pin down” the above (rather evasive) morphism and use it as an instrument in the study of group Γ(g, n). Thus our goal is two-fold: (i) to define a functor between Rie... |

6 | On complex and noncommutative
- Nikolaev
- 2006
(Show Context)
Citation Context ...aces and abelian groups with order; (ii) to apply the functor to open problems of conformal geometry, e.g. Harvey’s Conjecture. As far as question (i), we established earlier such a functor (see e.g. =-=[19]-=-) but give a full treatment here for reader’s comfort. Namely, each conformal structure either on the torus or any higher genus surface can be assigned a canonical ordered abelian group G (dimension g... |

4 |
Zur Theorie der konformen Abbildung, Göttinger Nachrichten
- Hilbert
- 1909
(Show Context)
Citation Context ...uniformization of Riemann surfaces There exists a canonical way to represent Riemann surfaces by the interval exchange transformations, see e.g. Bödigheimer [1]. The construction goes back to Hilbert =-=[12]-=-. Let us remind the main ideas, see [2] and [12]. Roughly, every Riemann surface S admits a conformal mapping to the complex plane with a finite set of the parallel half-lines (rays) deleted. Those ra... |

3 |
Interval exchange spaces and moduli spaces
- Bödigheimer
- 1993
(Show Context)
Citation Context ... Appendix for the definition of such groups). Connection between the Riemann surfaces, quadratic differentials and interval exchange transformation has been discussed by many authors, see Bödigheimer =-=[1]-=-, Masur [18] and Veech [22]. First we remind the reader how this connection comes in, and next we use it to define dimension groups associated to Riemann surfaces. 1.1 Interval exchange transformation... |

3 | A note on invariants of flows induced by Abelian differentials - Eckl |