×

Torsion in \(K\)-theory for boundary actions on affine buildings of type \(\widetilde{A}_n\). (English) Zbl 0980.46052

Let \(\Delta\) be a locally affine building of type \(\widetilde{A}_n\), \(n\in{\mathbb N}\), i.e., an \(n\)-dimensional simplicial complex and let \(\Delta^0\) be its vertex set. There is a type map \(\tau:\Delta^0\to{\mathbb Z}/(n+1){\mathbb Z}\) such that each simplex of maximal dimension (chamber) has exactly one vertex of each type. An automorphism \(\alpha\) of \(\Delta\) is type-rotating if there exists \(k\in{\mathbb Z}/(n+1){\mathbb Z}\) such that \(\tau(\alpha v)=\tau(v)+k\) for all \(v\in\Delta^0\). An apartment in \(\Delta\) is a subcomplex that is isomorphic to a Coxeter complex of type \(\widetilde{A}_n\) and a sector is a simplicial cone made up of chambers in some apartment and containing a unique base chamber. Two sectors are equivalent if their intersection contains a sector. The boundary \(\partial\Delta\) of \(\Delta\) is the set of equivalence classes of sectors in \(\Delta\).
Let \(\Gamma\) be a torsion-free discrete group of type rotating automorphisms of \(\Delta\) and let \([I]\) denote the class of the unit \(I\) of the \(C^*\)-algebra \(C(\partial\Delta)\rtimes\Gamma\) in the \(K\)-theory group \(K_0(C(\partial\Delta)\rtimes \Gamma)\). The main result of the paper is that \([I]\) is a torsion element if \(\Gamma\) acts cocompactly on \(\Delta\). An explicit bound for the order of \([I]\) is given and it is shown that for \(n=1,2\) the Euler-Poincaré characteristic \(\chi(\Gamma)\) annihilates \([I]\).

MSC:

46L80 \(K\)-theory and operator algebras (including cyclic theory)
51E24 Buildings and the geometry of diagrams
20G25 Linear algebraic groups over local fields and their integers
58B34 Noncommutative geometry (à la Connes)
46L55 Noncommutative dynamical systems
PDFBibTeX XMLCite
Full Text: DOI arXiv