Aperiodic tilings
The author describes a construction of certain \(X-\)tiling systems. In particular he obtains an aperiodic \(X-\)tiling system for \(X=G/K\) where \(G\) is a semisimple Lie group without compact factors and different from SL\(_2(\mathbb{R}).\) The construction is based on the use of two non-commensurable uniform lattices in \(G\) combined with Mostow rigidity. In the case of \(X=G/K\) with \(G\) being either Sp\((n,1), n\geqq 2\) (here \(X\) is a quaternionic hyperbolic space \(\mathbb{H}\mathbb{H}^n\)), \(F_4^{-20}\) (here \(X\) is the Cayley hyperbolic plane \(\mathbb{C}\)a\(\mathbb{C}^2\)) or a noncompact simple Lie group of rank at least two, a strongly aperiodic tiling system is constructed. The construction is based on the use of two uniform lattices which are disjoint even after arbitrary conjugations and the rigidity results of \textit{P. Pansu} [Ann. Math., II. Ser. 129, No. 1, 1-60 (1989; Zbl 0678.53042)] and \textit{B. Kleiner} and \textit{B. Leeb} [`Rigidity of quasiisometries for symmetric spaces of higher rank', preprint].
linear algebraic groups over the quaternions
scientific article; zbMATH DE number 1028193