Torsion in boundary coinvariants and \(K\)-theory for affine buildings (Q2487346)

The author deals with coinvariants for group actions on the boundary of an affine building and studies the connection with the \(K\)-theory of the boundary algebra \({\mathcal A}_\Gamma\). He shows that if \((G,I,N,S)\) is an affine topological Tits system of rank \(n+1\) with associated Tits complex \(\Delta\) and boundary \(\Omega\) and \(\Gamma\) is a torsion-free (cocompact) lattice in \(G\), then the module of \(\Gamma\)-coinvariants \(\Omega_\Gamma=H_0(\Gamma;C(\Omega,Z))\) is a finitely generated abelian group such that (1) the class [\textbf{1}] of the identity function \textbf{1} in \(C(\Omega,Z)\) has finite order in \(\Omega_\Gamma\); (2) the order of [\textbf{1}] in \(\Omega_\Gamma\) satisfies ord([\textbf{1}])\(<q_s\cdot\text{ covol}(\Gamma)\) where \(s\in I\) is a special type and \(q_s+1\) is the number of chambers in \(\Delta\) which contain a given simplex of dimension \(n-1\) and type \(I-\{s\}\); (3) ord([\textbf{1}])\(<\text{ covol}(\Gamma)\) if \(G\) is not one of the exceptional types \(\widetilde G_2\), \(\widetilde F_4\) or \(\widetilde{E}_8\). Improved estimates are obtained in the rank 2 case with some exact computational results for buildings of type \(\widetilde A_2\); compare the author [K-Theory 22, 251--269 (2001; Zbl 0980.46052)] and the author and \textit{T. Steger} [Can. J. Math. 53, 809--833 (2001; Zbl 0993.46039)]. As an application of this result to algebraic groups the author obtains that the class [\textbf{1}] has torsion in \(\Omega_\Gamma\) if \(\Gamma\) is a lattice in the group \(G\) of \(k\)-rational points of an absolutely almost simple, simply connected linear algebraic \(k\)-group, where \(k\) is a non-archimedian local field of characteristic zero. A similar consequence applies to the \(K_0\) group of the boundary crossed product \(C^*\)-algebra \({\mathcal A}_\Gamma=C(\Omega)\Gamma\) in that \([{\mathbf 1}]_{K_0}\) has finite order in \(K_0({\mathcal A}_\Gamma)\).