On the Haagerup inequality and groups acting on A ˜ n -buildings

Alain Valette

Annales de l'institut Fourier (1997)

  • Volume: 47, Issue: 4, page 1195-1208
  • ISSN: 0373-0956

Abstract

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Let Γ be a group endowed with a length function L , and let E be a linear subspace of C Γ . We say that E satisfies the Haagerup inequality if there exists constants C , s > 0 such that, for any f ∈ E , the convolutor norm of f on ℓ 2 ( Γ ) is dominated by C times the ℓ 2 norm of f ( 1 + L ) s . We show that, for E = C Γ , the Haagerup inequality can be expressed in terms of decay of random walks associated with finitely supported symmetric probability measures on Γ . If L is a word length function on a finitely generated group Γ , we show that, if the space Rad L ( Γ ) of radial functions with respect to L satisfies the Haagerup inequality, then Γ is non-amenable if and only if Γ has superpolynomial growth. We also show that the Haagerup inequality for Rad L ( Γ ) has a purely combinatorial interpretation; thus, using the main result of the companion paper by J. Swiatkowski, we deduce that, for a group Γ acting simply transitively on the vertices of a thick euclidean building of type A ˜ n , the space Rad L ( Γ ) satisfies the Haagerup inequality, and Γ is non-amenable.

How to cite

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Valette, Alain. "On the Haagerup inequality and groups acting on $\tilde{A}_n$-buildings." Annales de l'institut Fourier 47.4 (1997): 1195-1208. <http://eudml.org/doc/75259>.

@article{Valette1997,
abstract = {Let $\Gamma $ be a group endowed with a length function $L$, and let $E$ be a linear subspace of $\{\bf C\}\Gamma $. We say that $E$ satisfies the Haagerup inequality if there exists constants $C,s&gt;0$ such that, for any $f\in E$, the convolutor norm of $f$ on $\ell ^\{2\}(\Gamma )$ is dominated by $C$ times the $\ell ^\{2\}$ norm of $f(1+L)^\{s\}$. We show that, for $E=\{\bf C\}\Gamma $, the Haagerup inequality can be expressed in terms of decay of random walks associated with finitely supported symmetric probability measures on $\Gamma $. If $L$ is a word length function on a finitely generated group $\Gamma $, we show that, if the space $\{\rm Rad\}_\{L\}(\Gamma )$ of radial functions with respect to $L$ satisfies the Haagerup inequality, then $\Gamma $ is non-amenable if and only if $\Gamma $ has superpolynomial growth. We also show that the Haagerup inequality for $\{\rm Rad\}_\{L\}(\Gamma )$ has a purely combinatorial interpretation; thus, using the main result of the companion paper by J. Swiatkowski, we deduce that, for a group $\Gamma $ acting simply transitively on the vertices of a thick euclidean building of type $\tilde\{A\}_\{n\}$, the space $\{\rm Rad\}_\{L\}(\Gamma )$ satisfies the Haagerup inequality, and $\Gamma $ is non-amenable.},
author = {Valette, Alain},
journal = {Annales de l'institut Fourier},
keywords = {convolutor norm; random walks; amenability; growth of groups; euclidean buildings},
language = {eng},
number = {4},
pages = {1195-1208},
publisher = {Association des Annales de l'Institut Fourier},
title = {On the Haagerup inequality and groups acting on $\tilde\{A\}_n$-buildings},
url = {http://eudml.org/doc/75259},
volume = {47},
year = {1997},
}

TY - JOUR
AU - Valette, Alain
TI - On the Haagerup inequality and groups acting on $\tilde{A}_n$-buildings
JO - Annales de l'institut Fourier
PY - 1997
PB - Association des Annales de l'Institut Fourier
VL - 47
IS - 4
SP - 1195
EP - 1208
AB - Let $\Gamma $ be a group endowed with a length function $L$, and let $E$ be a linear subspace of ${\bf C}\Gamma $. We say that $E$ satisfies the Haagerup inequality if there exists constants $C,s&gt;0$ such that, for any $f\in E$, the convolutor norm of $f$ on $\ell ^{2}(\Gamma )$ is dominated by $C$ times the $\ell ^{2}$ norm of $f(1+L)^{s}$. We show that, for $E={\bf C}\Gamma $, the Haagerup inequality can be expressed in terms of decay of random walks associated with finitely supported symmetric probability measures on $\Gamma $. If $L$ is a word length function on a finitely generated group $\Gamma $, we show that, if the space ${\rm Rad}_{L}(\Gamma )$ of radial functions with respect to $L$ satisfies the Haagerup inequality, then $\Gamma $ is non-amenable if and only if $\Gamma $ has superpolynomial growth. We also show that the Haagerup inequality for ${\rm Rad}_{L}(\Gamma )$ has a purely combinatorial interpretation; thus, using the main result of the companion paper by J. Swiatkowski, we deduce that, for a group $\Gamma $ acting simply transitively on the vertices of a thick euclidean building of type $\tilde{A}_{n}$, the space ${\rm Rad}_{L}(\Gamma )$ satisfies the Haagerup inequality, and $\Gamma $ is non-amenable.
LA - eng
KW - convolutor norm; random walks; amenability; growth of groups; euclidean buildings
UR - http://eudml.org/doc/75259
ER -

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