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Asymptotic $K$-Theory for Groups Acting on $\tA_2$ Buildings

  Published:2001-08-01
 Printed: Aug 2001
  • Guyan Robertson
  • Tim Steger
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Abstract

Let $\Gamma$ be a torsion free lattice in $G=\PGL(3, \mathbb{F})$ where $\mathbb{F}$ is a nonarchimedean local field. Then $\Gamma$ acts freely on the affine Bruhat-Tits building $\mathcal{B}$ of $G$ and there is an induced action on the boundary $\Omega$ of $\mathcal{B}$. The crossed product $C^*$-algebra $\mathcal{A}(\Gamma)=C(\Omega) \rtimes \Gamma$ depends only on $\Gamma$ and is classified by its $K$-theory. This article shows how to compute the $K$-theory of $\mathcal{A}(\Gamma)$ and of the larger class of rank two Cuntz-Krieger algebras.
Keywords: $K$-theory, $C^*$-algebra, affine building $K$-theory, $C^*$-algebra, affine building
MSC Classifications: 46L80, 51E24 show english descriptions $K$-theory and operator algebras (including cyclic theory) [See also 18F25, 19Kxx, 46M20, 55Rxx, 58J22]
Buildings and the geometry of diagrams
46L80 - $K$-theory and operator algebras (including cyclic theory) [See also 18F25, 19Kxx, 46M20, 55Rxx, 58J22]
51E24 - Buildings and the geometry of diagrams
 

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