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A HAAGERUP INEQUALITY, DEFORMATION OF TRIANGLES AND AFFINE BUILDINGS

Published online by Cambridge University Press:  18 January 2006

Malik Talbi
Affiliation:
Westfälische Wilhelms-Universität Münster, Mathematisches Institut, Einsteinstraβe 62, 48149 Münster, Germany ([email protected]) Present address: Institut Camille Jordan, Université Claude Bernard Lyon 1, 43 boulevard du 11 novembre 1918, 69622 Villeurbanne cedex, France ([email protected])

Abstract

In this paper we study a Haagerup inequality in the general case of discrete groupoids. We develop two geometrical tools, pinching and tetrahedral change of faces, based on deformation of triangles, to prove it. We show how to use these tools to find all the already known results just by manipulating triangles. We use these tools for groups acting freely and by isometries on the set of vertices of any affine building and give a first reduction of this inequality to its verification on some special triangles and prove the inequality when the building is of type $\tilde{A}_{k_1}\times\cdots\times\tilde{A}_{k_n}$, where $k_i\in\{1,2\}$, $i=1,\dots,n$.

Type
Research Article
Copyright
2006 Cambridge University Press

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