\(\Sigma\)-chamber systems, coloured graphs and orbifolds (Q1922674)

Two approaches to the problem of encoding manifolds by means of combinatorial structures are considered. One of these uses coloured graphs and the other \(\Sigma\)-chambers which are tilings of a simply connected manifold on which a group acts. Elements of this group preserve the invariants of these tilings. Some notions on the second approach are recalled from [\textit{A. W. M. Dress}, Adv. Math. 63, 196-212 (1987; Zbl 0614.57024)]. It is stated that \(\Sigma\)-chamber systems correspond to coloured graphs. For a fixed group \(G\), a \(G\)-set is defined to be a set \(M\) together with a right action of \(G\). Depending on this, \(G\)-coverings are defined as morphisms between \(G\)-sets, and results concerning coverings and universal coverings are carried to \(G\)-sets. Delaney symbols are recalled, in a different manner, by means of \(\Sigma\)-sets, and relations between these symbols and \(\Sigma\)-coverings are discussed. Finally, a Delaney symbol is associated to every orbifold.