Groupoids and Noncommutative Geometry Seminar 

21/05/2007, 15:00 — 16:00 — Room P3.10, Mathematics Building
Valentin Deaconu, University of Nevada, Reno

Groupoids associated to a textile system

The notion of textile system was introduced by M. Nasu in order to analyze endomorphisms of topological Markov chains. It consists of two graphs $G$ and $H$ and two morphisms $p,q:G\to H$, with some extra properties. In the case $p$ and $q$ have the path lifting property, we prove that they induce groupoid morphisms $\pi ,\rho :\Gamma (G)\to \Gamma (H)$ between the corresponding étale groupoids of $G$ and $H$. This way, $C^{*}(G)$ is the C*-algebra of two different Fell bundles over $\Gamma (H)$. It turns out that a textile system determines a first quadrant two-dimensional subshift of finite type, via a collection of Wang tiles, and conversely, any such subshift is conjugate to a textile shift. Our groupoid morphisms and C*-algebras encode the complexity of these two-dimensional subshifts. Several concrete examples will be considered.