Relative property (T) and linear groups

Talia Fernós[1]

  • [1] University of Illinois at Chicago Dept. of MSCS (m/c 249) 851 South Morgan Street Chicago, IL 60607-7045 (USA)

Annales de l’institut Fourier (2006)

  • Volume: 56, Issue: 6, page 1767-1804
  • ISSN: 0373-0956

Abstract

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Relative property (T) has recently been used to show the existence of a variety of new rigidity phenomena, for example in von Neumann algebras and the study of orbit-equivalence relations. However, until recently there were few examples of group pairs with relative property (T) available through the literature. This motivated the following result: A finitely generated group Γ admits a special linear representation with non-amenable R -Zariski closure if and only if it acts on an Abelian group A (of finite nonzero Q -rank) so that the corresponding group pair ( Γ ⋉ A , A ) has relative property (T).The proof is constructive. The main ingredients are Furstenberg’s celebrated lemma about invariant measures on projective spaces and the spectral theorem for the decomposition of unitary representations of Abelian groups. Methods from algebraic group theory, such as the restriction of scalars functor, are also employed.

How to cite

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Fernós, Talia. "Relative property (T) and linear groups." Annales de l’institut Fourier 56.6 (2006): 1767-1804. <http://eudml.org/doc/10191>.

@article{Fernós2006,
abstract = {Relative property (T) has recently been used to show the existence of a variety of new rigidity phenomena, for example in von Neumann algebras and the study of orbit-equivalence relations. However, until recently there were few examples of group pairs with relative property (T) available through the literature. This motivated the following result: A finitely generated group $\Gamma $ admits a special linear representation with non-amenable $R$-Zariski closure if and only if it acts on an Abelian group $A$ (of finite nonzero $Q$-rank) so that the corresponding group pair $(\Gamma \ltimesA,A)$ has relative property (T).The proof is constructive. The main ingredients are Furstenberg’s celebrated lemma about invariant measures on projective spaces and the spectral theorem for the decomposition of unitary representations of Abelian groups. Methods from algebraic group theory, such as the restriction of scalars functor, are also employed.},
affiliation = {University of Illinois at Chicago Dept. of MSCS (m/c 249) 851 South Morgan Street Chicago, IL 60607-7045 (USA)},
author = {Fernós, Talia},
journal = {Annales de l’institut Fourier},
keywords = {Relative property (T); group extensions; linear algebraic groups; relative property (T)},
language = {eng},
number = {6},
pages = {1767-1804},
publisher = {Association des Annales de l’institut Fourier},
title = {Relative property (T) and linear groups},
url = {http://eudml.org/doc/10191},
volume = {56},
year = {2006},
}

TY - JOUR
AU - Fernós, Talia
TI - Relative property (T) and linear groups
JO - Annales de l’institut Fourier
PY - 2006
PB - Association des Annales de l’institut Fourier
VL - 56
IS - 6
SP - 1767
EP - 1804
AB - Relative property (T) has recently been used to show the existence of a variety of new rigidity phenomena, for example in von Neumann algebras and the study of orbit-equivalence relations. However, until recently there were few examples of group pairs with relative property (T) available through the literature. This motivated the following result: A finitely generated group $\Gamma $ admits a special linear representation with non-amenable $R$-Zariski closure if and only if it acts on an Abelian group $A$ (of finite nonzero $Q$-rank) so that the corresponding group pair $(\Gamma \ltimesA,A)$ has relative property (T).The proof is constructive. The main ingredients are Furstenberg’s celebrated lemma about invariant measures on projective spaces and the spectral theorem for the decomposition of unitary representations of Abelian groups. Methods from algebraic group theory, such as the restriction of scalars functor, are also employed.
LA - eng
KW - Relative property (T); group extensions; linear algebraic groups; relative property (T)
UR - http://eudml.org/doc/10191
ER -

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