Nariya Kawazumi

Periodic surface automorphisms and algebraic independence of Morita–Mumford classes

Abstract We prove a vanishing theorem for the Morita–Mumford classes on periodic surface automorphisms, and construct enough periodic automorphisms to give an alternative and elementary proof of the stable rational algebraic independence of the Morita–Mumford classes, originally shown by Miller (J. Differential Geom. 24 (1986) 1–14) and Morita (Invent. Math. 90 (1987) 551–557).

Canonical 2-forms on the moduli space of Riemann surfaces

A summary introduction of the Weil-Petersson metric space geometry is presented. Teichmueller space and its augmentation are described in terms of Fenchel-Nielsen coordinates. Formulas for the gradients and Hessians of geodesic-length functions are presented. Applications are considered. A description of the Weil-Petersson metric in Fenchel-Nielsen coordinates is presented. The Alexandrov tangent cone at points of the augmentation is described. A comparison dictionary is presented between the geometry of the space of flat tori and Teichmueller space with the Weil-Petersson metric.This survey paper begins with the description of the duality between arc systems and ribbon graphs embedded in a punctured surface. Then we explain how to cellularize the moduli space of curves in two different ways: using Jenkins-Strebel differentials and using hyperbolic geometry. We also briefly discuss how these two methods are related. Next, we recall the definition of Witten cycles and we illustrate their connection with tautological classes and Weil-Petersson geometry. Finally, we exhibit a simple direct argument to prove that Witten classes are stable.This is a survey of the theory of complex projective (CP^1) structures on compact surfaces. After some preliminary discussion and definitions, we concentrate on three main topics: (1) Using the Schwarzian derivative to parameterize the moduli space (2) Thurstons parameterization of the moduli space using grafting (3) Holonomy representations of CP^1 structures We also discuss some results comparing the two parameterizations of the space of projective structures and relating these parameterizations to the holonomy map.The conjugacy class of a generic unimodular 2 by 2 complex matrix is determined by its trace, which may be an arbitrary complex number. In the nineteenth century, it was known that a generic pair (X,Y) of such pairs is determined up to conjugacy by the triple of traces (tr(X),tr(Y),tr(XY), which may be an arbitary element of C^3. This paper gives an elementary and detailed proof of this fact, which was published by Vogt in 1889. The folk theorem describing the extension of a representation to a representation of the index-two supergroup which is a free product of three groups of order two, is described in detail, and related to hyperbolic geometry. When n > 2, the classification of conjugacy-classes of n-tuples in SL(2,C) is more complicated. We describe it in detail when n= 3. The deformation spaces of hyperbolic structures on some simple surfaces S whose fundamental group is free of rank two or three are computed in trace coordinates. (We only consider the two orientable surfaces whose fundamental group has rank 3.)This article is a survey on the braid groups, the Artin groups, and the Garside groups. It is a presentation, accessible to non-experts, of various topological and algebraic aspects of these groups. It is also a report on three points of the theory: the faithful linear representations, the cohomology, and the geometrical representations.Denote the free group on 2 letters by F_2 and the SL(2,C)-representation variety of F_2 by R=Hom(F_2,SL(2,C)). The group SL(2,C) acts on R by conjugation. We construct an isomorphism between the coordinate ring C[SL(2,C)] and the ring of matrix coefficients, providing an additive basis of C[R]^SL(2,C) in terms of spin networks. Using a graphical calculus, we determine the symmetries and multiplicative structure of this basis. This gives a canonical description of the regular functions on the SL(2,C)-character variety of F_2 and a new proof of a classical result of Fricke, Klein, and Vogt.The article under review is a concise but contemporary survey of infinite-dimensional Teichmuller spaces. In particular, it contains recent remarkable results by the authors on this subject.

Twisted Morita-Mumford classes on braid groups

Evaluating the twisted Morita-Mumford classes bar(h)_p on the Artin braid group B_n, we give the stable algebraic independence of the bar(h)_ps on the automorphism group of the free group, Aut(F_n). This is sharper than the results we obtained by restricting them to the mapping class group.

Weierstrass points and Morita–Mumford classes on hyperelliptic mapping class groups

Abstract We describe the Morita–Mumford classes on the hyperelliptic mapping class group of genus g ⩾2, and prove a slightly weakened version of Akitas conjectures in the hyperelliptic case. The proof involves the notion of Weierstrass points.

Generators of the homological Goldman Lie algebra

We determine the minimal number of generators of the homological Goldman Lie algebra of a surface consisting of elements of the first homology group of the surface.

Integral Riemann–Roch formulae for cyclic subgroups of mapping class groups

The first author conjectured certain relations for Morita–Mumford classes and Newton classes in the integral cohomology of mapping class groups (integral Riemann–Roch formulae). In this paper, the conjecture is verified for cyclic subgroups of mapping class groups.