A commentary on Teichmüller’s paper `Veränderliche Riemannsche Flächen'

A’Campo–Neuen, Annette; A'Campo, Norbert; Ji, Lizhen; Papadopoulos, Athanase

doi:10.4171/117-1/20

Cited by 11 publications

(48 citation statements)

References 12 publications

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“…The universal family is usually called the universal Teichmüller curve. For more details and an historical account, see the survey paper [1]. Proposition 7.8.…”

Section: 2mentioning

confidence: 99%

The geometry of maximal components of the PSp(4, ℝ) character variety

Alessandrini

Collier

2019

Geom. Topol.

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In this paper we describe the space of maximal components of the character variety of surface group representations into PSp(4, R) and Sp(4, R).For every real rank 2 Lie group of Hermitian type, we construct a mapping class group invariant complex structure on the maximal components. For the groups PSp(4, R) and Sp(4, R), we give a mapping class group invariant parameterization of each maximal component as an explicit holomorphic fiber bundle over Teichmüller space. Special attention is put on the connected components which are singular, we give a precise local description of the singularities and their geometric interpretation. We also describe the quotient of the maximal components of PSp(4, R) and Sp(4, R) by the action of the mapping class group as a holomorphic submersion over the moduli space of curves.These results are proven in two steps, first we use Higgs bundles to give a non-mapping class group equivariant parameterization, then we prove an analogue of Labourie's conjecture for maximal PSp(4, R) representations.The spaces X max,sw2 sw1 (Γ, PSp(4, R)) are singular, but the singularities consist only of Z 2 and Z 2 ⊕ Z 2 -orbifold points. The space H ′ /Z 2 also has an orbifold structure. The homeomorphism above is an orbifold isomorphism, in particular, it is smooth away from the singular set.Corollary 7. Each space X max,sw2 sw1 (Γ, PSp(4, R)) deformation retracts onto the quotient of (S 1 ) 2g−2 by inversion. In particular, its rational cohomology is: PSp(4, R)), Q) ∼ = H j ((S 1 ) 2g−2 , Q) if j is even, 0 otherwise. Character varieties and Higgs bundlesIn this section we recall general facts about character varieties and Higgs bundles.

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“…The universal family is usually called the universal Teichmüller curve. For more details and an historical account, see the survey paper [1]. Proposition 7.8.…”

Section: 2mentioning

confidence: 99%

The geometry of maximal components of the PSp(4, ℝ) character variety

Alessandrini

Collier

2019

Geom. Topol.

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“…These written texts were usually distributed as mimeographed notes at the subsequent seminar meetings. 2 Thus, we have at our disposal a series of papers by Grothendieck on Teichmüller theory ( [21] to [30]). We 1 Although this will seem obvious to everybody, we point out that all the occurrences of the name Cartan in this chapter refer to Henri Cartan.…”

Section: Introductionmentioning

confidence: 99%

“…(This will exempt us from adding the first initial H. each time we write his name, as is usually required in the mathematical literature, to distinguish him from his father Elie Cartan.) 2 Talking about Cartan and his seminar, Douady writes in [13]: "He did not tolerate the slightest inaccuracy, the slightest imprecision, and he criticized the speaker to the point of destabilizing him. [...] But what really mattered for Cartan were the notes of the exposés.…”

Section: Introductionmentioning

confidence: 99%

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On Grothendieck’s construction of Teichmüller space

A'Campo

Papadopoulos

2016

IRMA Lectures in Mathematics and Theoretical Physics

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Abstract. In his 1944 paper Veränderliche Riemannsche Flächen, Teichmüller defined a structure of complex manifold on the set of isomorphism classes of marked closed Riemann surfaces of genus g. The complex manifold he obtained is the space called today Teichmüller space. In the same paper, Teichmüller introduced the so-called universal Teichmüller curve -a space over Teichmüller space where the fiber above each point is a Riemann surface representing that point. In fact, Teichmüller proved the existence of the Teichmüller curve as a space of Riemann surfaces parametrized by an analytic space, with an existence and uniqueness theorem establishing this analytic structure. This result was later reformulated and proved by Grothendieck in a series of ten lectures he gave at Cartan's seminar in [1960][1961]. In his approach, Grothendieck replaced Teichmüller's explicit parameters by a general construction of fiber bundles whose base is an arbitrary analytic space. This work on Teichmüller space led him to recast the bases of analytic geometry using the language of categories and functors. In Grothendieck's words, the Teichmüller curve becomes a space representing a functor from the category of analytic spaces into the category of sets. In this survey, we comment on Grothendieck's series of lectures.The survey is primarily addressed to low-dimensional topologists and geometers. In presenting Grothendieck's results, we tried to explain or rephrase in more simple terms some notions that are usually expressed in the language of algebraic geometry. However, it is not possible to short-circuit the language of categories and functors.The survey is also addressed to those algebraic geometers who wish to know how the notion of moduli space evolved in connection with Teichmüller theory.Explaining the origins of mathematical ideas contributes in dispensing justice to their authors and it usually renders the theory that is surveyed more attractive.

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“…The Teichmüller tower can be thought of as the set of all algebraic fundamental groups of moduli spaces organized in a coherent way as finite covers of each other. 1 A nodal curve is a connected projective singular curve whose singularities, called nodes, that is, they are isolated with local formal model the two coordinate axes in the affine space A 2 in a neighborhood of the origin. For stable nodal curves, it is also required that the Euler characteristic of each of the curves which are connected components of the complement of the nodes in such a surface is negative.…”

Section: Introductionmentioning

confidence: 99%

Actions of the absolute Galois group

A'Campo¹,

Ji²,

Papadopoulos³

2016

IRMA Lectures in Mathematics and Theoretical Physics

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We review some ideas of Grothendieck and others on actions of the absolute Galois group Γ Q of Q (the automorphism group of the tower of finite extensions of Q), related to the geometry and topology of surfaces (mapping class groups, Teichmüller spaces and moduli spaces of Riemann surfaces). Grothendieck's motivation came in part from his desire to understand the absolute Galois group. But he was also interested in Thurston's work on surfaces, and he expressed this in his Esquisse d'un programme, his Récoltes et semailles and on other occasions. He introduced the notions of dessin d'enfant, Teichmüller tower, and other related objects, he considered the actions of Γ Q on them or on theirétale fundamental groups, and he made conjectures on some natural homomorphisms between the absolute Galois group and the automorphism groups (or outer automorphism groups) of these objects. We mention several ramifications of these ideas, due to various authors. We also report on the works of Sullivan and others on nonlinear actions of Γ Q , in particular in homotopy theory.

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A commentary on Teichmüller’s paper `Veränderliche Riemannsche Flächen'

Cited by 11 publications

References 12 publications

The geometry of maximal components of the PSp(4, ℝ) character variety

The geometry of maximal components of the PSp(4, ℝ) character variety

On Grothendieck’s construction of Teichmüller space

Actions of the absolute Galois group

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