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Cross ratios, surface groups, P S L ( n , 𝐑 ) and diffeomorphisms of the circle

François Labourie

Publications Mathématiques de l'IHÉS (2007)

  • Volume: 106, page 139-213
  • ISSN: 0073-8301

Abstract

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This article relates representations of surface groups to cross ratios. We first identify a connected component of the space of representations into PSL(n,ℝ) – known as the n-Hitchin component– to a subset of the set of cross ratios on the boundary at infinity of the group. Similarly, we study some representations into C 1 , h ( 𝕋 ) ⋊ Diff h ( 𝕋 ) associated to cross ratios and exhibit a “character variety” of these representations. We show that this character variety contains alln-Hitchin components as well as the set of negatively curved metrics on the surface.

How to cite

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Labourie, François. "Cross ratios, surface groups, $PSL(n,\mathbf {R})$ and diffeomorphisms of the circle." Publications Mathématiques de l'IHÉS 106 (2007): 139-213. <http://eudml.org/doc/104227>.

@article{Labourie2007,
abstract = {This article relates representations of surface groups to cross ratios. We first identify a connected component of the space of representations into PSL(n,ℝ) – known as the n-Hitchin component– to a subset of the set of cross ratios on the boundary at infinity of the group. Similarly, we study some representations into $C^\{1,h\}(\mathbb \{T\})\rtimes \text\{Diff\}^\{h\}(\mathbb \{T\})$ associated to cross ratios and exhibit a “character variety” of these representations. We show that this character variety contains alln-Hitchin components as well as the set of negatively curved metrics on the surface.},
author = {Labourie, François},
journal = {Publications Mathématiques de l'IHÉS},
keywords = {fundamental group representations; character variety; cross ratio; Hitchin component; higher Teichmüller theory},
language = {eng},
pages = {139-213},
publisher = {Springer},
title = {Cross ratios, surface groups, $PSL(n,\mathbf \{R\})$ and diffeomorphisms of the circle},
url = {http://eudml.org/doc/104227},
volume = {106},
year = {2007},
}

TY - JOUR
AU - Labourie, François
TI - Cross ratios, surface groups, $PSL(n,\mathbf {R})$ and diffeomorphisms of the circle
JO - Publications Mathématiques de l'IHÉS
PY - 2007
PB - Springer
VL - 106
SP - 139
EP - 213
AB - This article relates representations of surface groups to cross ratios. We first identify a connected component of the space of representations into PSL(n,ℝ) – known as the n-Hitchin component– to a subset of the set of cross ratios on the boundary at infinity of the group. Similarly, we study some representations into $C^{1,h}(\mathbb {T})\rtimes \text{Diff}^{h}(\mathbb {T})$ associated to cross ratios and exhibit a “character variety” of these representations. We show that this character variety contains alln-Hitchin components as well as the set of negatively curved metrics on the surface.
LA - eng
KW - fundamental group representations; character variety; cross ratio; Hitchin component; higher Teichmüller theory
UR - http://eudml.org/doc/104227
ER -

References

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