On monodromy of complex projective structures

Kapovich, Michael

doi:10.1007/bf01245182

Cited by 19 publications

(13 citation statements)

References 22 publications

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“…2 Note that the discussion in [Ka,§7.2] does not distinguish linear and projective monodromy representations.…”

Section: Equivalence Of Obstructionsmentioning

confidence: 99%

“…This theorem shows that on a fixed Riemann surface, if any sequence of quadratic differentials diverge, so must the conjugacy classes of corresponding monodromy representations. A brief outline of the proof was given in [Ka,§7.2]. 2 As before, R denotes a closed Riemann surface of genus exceeding one and Q(R) its space of holomorphic quadratic differentials.…”

Section: Equivalence Of Obstructionsmentioning

confidence: 99%

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The Monodromy Groups of Schwarzian Equations on Closed Riemann Surfaces

Gallo

Kapovich

Marden³

2000

The Annals of Mathematics

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137

224

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Let θ : π 1 (R) → PSL(2, C) be a homomorphism of the fundamental group of an oriented, closed surface R of genus exceeding one. We will establish the following theorem.Theorem. Necessary and sufficient for θ to be the monodromy representation associated with a complex projective stucture on R, either unbranched or with a single branch point of order 2, is that θ(π 1 (R)) be nonelementary. A branch point is required if and only if the representation θ does not lift to SL(2, C).

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“…2 Note that the discussion in [Ka,§7.2] does not distinguish linear and projective monodromy representations.…”

Section: Equivalence Of Obstructionsmentioning

confidence: 99%

Section: Equivalence Of Obstructionsmentioning

confidence: 99%

The Monodromy Groups of Schwarzian Equations on Closed Riemann Surfaces

Gallo

Kapovich

Marden³

2000

The Annals of Mathematics

Self Cite

137

224

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“…En dimension n = 1 , l'étude des structures projectives sur la surface de Riemann M = Γ \ H 1 C de genre g ≥ 2 est un problème classique [8].…”

Section: Exemples : 1 Structures Projectives Complexes Sur Les Variéunclassified

Un théorème de rigidité non-métrique pour les variétés localement symétriques hermitiennes

Klingler¹

2001

Commentarii Mathematici Helvetici

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Abstract. Let X be an irreducible Hermitian symmetric space of non-compact type of dimension greater than 1 and G be the group of biholomorphisms of X ; let M = Γ \ X be a quotient of X by a torsion-free discrete subgroup Γ of G such that M is of finite volume in the canonical metric. Then, due to the G -equivariant Borel embedding of X into its compact dual X c , the locally symmetric structure of M can be considered as a special kind of a ( G C , Xc ) -structure on M , a maximal atlas of Xc -valued charts with locally constant transition maps in the complexified group G C . By Mostow's rigidity theorem the locally symmetric structure of M is unique. We prove that the ( G C , Xc ) -structure of M is the unique one compatible with its complex structure. In the rank one case this result is due to Mok and Yeung. Mathematics Subject Classification (2000). 53C35, 32M15, 22E40.

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“…In this note we compute the function d(ρ) in the very special case of representations with the image in the subgroup of translations. The general case will be treated elsewhere, here we only note that in [GKM00] (see also [Kap95]) it was shown that for each representation ρ with nonelementary image 1 , d(ρ) ∈ {0, 1} equals the 2-nd Stiefel-Whitney class of ρ (mod 2).…”

Section: Introductionmentioning

confidence: 99%

Periods of abelian differentials and dynamics

Kapovich

2020

Dynamics: Topology and Numbers

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Given a closed oriented surface S of genus ≥ 3 we describe those cohomology classes χ ∈ H 1 (S, C) which appear as the period characters of abelian differentials for some choice of complex structure τ = τ (χ) on S consistent with the orientation. In other words, we describe the union τ ∈T (S)where T (S) is the Teichmüller space of S. The proof is based upon Ratner's solution of Raghunathan's conjecture.

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On monodromy of complex projective structures

Abstract: Summary.We prove that for any nonelementary representation p: nl(S)-, SL(2, ~) of the fundamental group of a closed orientable hyperbolic surface S there exists a complex projective structure on S with the monodromy p.

Cited by 19 publications

References 22 publications

The Monodromy Groups of Schwarzian Equations on Closed Riemann Surfaces

The Monodromy Groups of Schwarzian Equations on Closed Riemann Surfaces

Un théorème de rigidité non-métrique pour les variétés localement symétriques hermitiennes

Periods of abelian differentials and dynamics

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