Constructing equivariant maps for representations

Stefano Francaviglia[1]

  • [1] Dipartimento di Matematica Applicata “U.Dini” via Buonarroti 1/c 56127 Pisa (Italy)

Annales de l’institut Fourier (2009)

  • Volume: 59, Issue: 1, page 393-428
  • ISSN: 0373-0956

Abstract

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We show that if Γ is a discrete subgroup of the group of the isometries of ℍ k , and if ρ is a representation of Γ into the group of the isometries of ℍ n , then any ρ -equivariant map F : ℍ k → ℍ n extends to the boundary in a weak sense in the setting of Borel measures. As a consequence of this fact, we obtain an extension of a result of Besson, Courtois and Gallot about the existence of volume non-increasing, equivariant maps. Then, we show that the weak extension we obtain is actually a measurable ρ -equivariant map in the classical sense. We use this fact to obtain measurable versions of Cannon-Thurston-type results for equivariant Peano curves. For example, we prove that if Γ is of divergence type and ρ is non-elementary, then there exists a measurable map D : ∂ ℍ k → ∂ ℍ n conjugating the actions of Γ and ρ ( Γ ) . Related applications are discussed.

How to cite

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Francaviglia, Stefano. "Constructing equivariant maps for representations." Annales de l’institut Fourier 59.1 (2009): 393-428. <http://eudml.org/doc/10397>.

@article{Francaviglia2009,
abstract = {We show that if $\Gamma $ is a discrete subgroup of the group of the isometries of $\mathbb\{H\}^k$, and if $\rho $ is a representation of $\Gamma $ into the group of the isometries of $\mathbb\{H\}^n$, then any $\rho $-equivariant map $F:\mathbb\{H\}^k \rightarrow \mathbb\{H\}^n$ extends to the boundary in a weak sense in the setting of Borel measures. As a consequence of this fact, we obtain an extension of a result of Besson, Courtois and Gallot about the existence of volume non-increasing, equivariant maps. Then, we show that the weak extension we obtain is actually a measurable $\rho $-equivariant map in the classical sense. We use this fact to obtain measurable versions of Cannon-Thurston-type results for equivariant Peano curves. For example, we prove that if $\Gamma $ is of divergence type and $\rho $ is non-elementary, then there exists a measurable map $D:\partial \mathbb\{H\}^k\rightarrow \partial \mathbb\{H\}^n$ conjugating the actions of $\Gamma $ and $\rho (\Gamma )$. Related applications are discussed.},
affiliation = {Dipartimento di Matematica Applicata “U.Dini” via Buonarroti 1/c 56127 Pisa (Italy)},
author = {Francaviglia, Stefano},
journal = {Annales de l’institut Fourier},
keywords = {Hyperbolic spaces; discrete groups; isometries; representation; equivariant; barycenter; natural map; volume; hyperbolic spaces},
language = {eng},
number = {1},
pages = {393-428},
publisher = {Association des Annales de l’institut Fourier},
title = {Constructing equivariant maps for representations},
url = {http://eudml.org/doc/10397},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Francaviglia, Stefano
TI - Constructing equivariant maps for representations
JO - Annales de l’institut Fourier
PY - 2009
PB - Association des Annales de l’institut Fourier
VL - 59
IS - 1
SP - 393
EP - 428
AB - We show that if $\Gamma $ is a discrete subgroup of the group of the isometries of $\mathbb{H}^k$, and if $\rho $ is a representation of $\Gamma $ into the group of the isometries of $\mathbb{H}^n$, then any $\rho $-equivariant map $F:\mathbb{H}^k \rightarrow \mathbb{H}^n$ extends to the boundary in a weak sense in the setting of Borel measures. As a consequence of this fact, we obtain an extension of a result of Besson, Courtois and Gallot about the existence of volume non-increasing, equivariant maps. Then, we show that the weak extension we obtain is actually a measurable $\rho $-equivariant map in the classical sense. We use this fact to obtain measurable versions of Cannon-Thurston-type results for equivariant Peano curves. For example, we prove that if $\Gamma $ is of divergence type and $\rho $ is non-elementary, then there exists a measurable map $D:\partial \mathbb{H}^k\rightarrow \partial \mathbb{H}^n$ conjugating the actions of $\Gamma $ and $\rho (\Gamma )$. Related applications are discussed.
LA - eng
KW - Hyperbolic spaces; discrete groups; isometries; representation; equivariant; barycenter; natural map; volume; hyperbolic spaces
UR - http://eudml.org/doc/10397
ER -

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