Signierte Zellenzerlegungen. II. (Signed cell decompositions. II.) (Q1820431)

\(\{\) p,q\(\}\) denotes a decomposition of the 2-sphere \((<)\), euclidean \((=)\) or hyperbolic \((>)\) plane (according to whether (p-1)(q-1) is \(<4\), \(=4\) or \(>4)\) into regular polygons where each polygon is bounded by p edges and at every vertex there meet q edges. The aim is to find necessary and sufficient conditions for a path \((a_ 0a_ 1...a_ na_{n+1})\) to be closed; here the \(a_ i\) are vertices forming the i- th segment \(a_ ia_{i+1}\) of the path. This is done by using the ''angle measures'' \(2\pi w_ i/q\) between the (i-1)-st and i-th segment, \(i=1,...,n\), of the path. If \(\alpha\) and \(\beta\) are the usual generators of the group of orientation preserving lattice automorphisms (i.e. \(\alpha^ 2=\beta^ p=(\alpha \beta)^ q=e=identity)\) then the path is closed if and only if \(\beta_{w_ 1}\beta_{w_ 2}...\beta_{w_ n}=e\) where \(\beta_ u=\alpha (\alpha \beta)^ u\). A similar result is obtained for more general sequences of exponents. For part I of this paper see Acta. Math. Acad. Sci. Hung. 22, 51-63 (1971; Zbl 0222.05104).