Harmonic maps and representations of non-uniform lattices of PU ( m , 1 )

Vincent Koziarz[1]; Julien Maubon[1]

  • [1] Université Henri Poincaré Institut Elie Cartan BP 239 54506 Vandœuvre-lès-Nancy Cedex (France)

Annales de l’institut Fourier (2008)

  • Volume: 58, Issue: 2, page 507-558
  • ISSN: 0373-0956

Abstract

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We study representations of lattices of PU ( m , 1 ) into PU ( n , 1 ) . We show that if a representation is reductive and if m is at least 2, then there exists a finite energy harmonic equivariant map from complex hyperbolic m -space to complex hyperbolic n -space. This allows us to give a differential geometric proof of rigidity results obtained by M. Burger and A. Iozzi. We also define a new invariant associated to representations into PU ( n , 1 ) of non-uniform lattices in PU ( 1 , 1 ) , and more generally of fundamental groups of orientable surfaces of finite topological type and negative Euler characteristic. We prove that this invariant is bounded by a constant depending only on the Euler characteristic of the surface and we give a complete characterization of representations with maximal invariant, thus generalizing the results of D. Toledo for uniform lattices.

How to cite

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Koziarz, Vincent, and Maubon, Julien. "Harmonic maps and representations of non-uniform lattices of ${\rm PU}(m,1)$." Annales de l’institut Fourier 58.2 (2008): 507-558. <http://eudml.org/doc/10323>.

@article{Koziarz2008,
abstract = {We study representations of lattices of $\{\rm PU\}(m,1)$ into $\{\rm PU\}(n,1)$. We show that if a representation is reductive and if $m$ is at least 2, then there exists a finite energy harmonic equivariant map from complex hyperbolic $m$-space to complex hyperbolic $n$-space. This allows us to give a differential geometric proof of rigidity results obtained by M. Burger and A. Iozzi. We also define a new invariant associated to representations into $\{\rm PU\}(n,1)$ of non-uniform lattices in $\{\rm PU\}(1,1)$, and more generally of fundamental groups of orientable surfaces of finite topological type and negative Euler characteristic. We prove that this invariant is bounded by a constant depending only on the Euler characteristic of the surface and we give a complete characterization of representations with maximal invariant, thus generalizing the results of D. Toledo for uniform lattices.},
affiliation = {Université Henri Poincaré Institut Elie Cartan BP 239 54506 Vandœuvre-lès-Nancy Cedex (France); Université Henri Poincaré Institut Elie Cartan BP 239 54506 Vandœuvre-lès-Nancy Cedex (France)},
author = {Koziarz, Vincent, Maubon, Julien},
journal = {Annales de l’institut Fourier},
keywords = {Representations; non-uniform lattices; complex hyperbolic space; Toledo invariant; harmonic maps; surfaces of finite topological type; rigidity; representations; Toledo invariants},
language = {eng},
number = {2},
pages = {507-558},
publisher = {Association des Annales de l’institut Fourier},
title = {Harmonic maps and representations of non-uniform lattices of $\{\rm PU\}(m,1)$},
url = {http://eudml.org/doc/10323},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Koziarz, Vincent
AU - Maubon, Julien
TI - Harmonic maps and representations of non-uniform lattices of ${\rm PU}(m,1)$
JO - Annales de l’institut Fourier
PY - 2008
PB - Association des Annales de l’institut Fourier
VL - 58
IS - 2
SP - 507
EP - 558
AB - We study representations of lattices of ${\rm PU}(m,1)$ into ${\rm PU}(n,1)$. We show that if a representation is reductive and if $m$ is at least 2, then there exists a finite energy harmonic equivariant map from complex hyperbolic $m$-space to complex hyperbolic $n$-space. This allows us to give a differential geometric proof of rigidity results obtained by M. Burger and A. Iozzi. We also define a new invariant associated to representations into ${\rm PU}(n,1)$ of non-uniform lattices in ${\rm PU}(1,1)$, and more generally of fundamental groups of orientable surfaces of finite topological type and negative Euler characteristic. We prove that this invariant is bounded by a constant depending only on the Euler characteristic of the surface and we give a complete characterization of representations with maximal invariant, thus generalizing the results of D. Toledo for uniform lattices.
LA - eng
KW - Representations; non-uniform lattices; complex hyperbolic space; Toledo invariant; harmonic maps; surfaces of finite topological type; rigidity; representations; Toledo invariants
UR - http://eudml.org/doc/10323
ER -

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