Combinatorial classification of fundamental domains of finite area for planar discontinuous isometry groups (Q920444)

Let G be a finitely generated discontinuous group acting on \(\Pi\) where \(\Pi\) is the Euclidean plane or the 2-sphere or the hyperbolic plane. Using a method introduced in a previous paper the authors find estimates \(n_{\min}\) and \(n_{\max}\) for the number n of vertices of a fundamental domain F of finite area for G. Here a point v of \(\Pi\) is called a vertex of F if it belongs to three distinct tiles F, g(F), h(F) where g,h\(\in G\), or if v lies on an edge of F and is the center of a point-reflection in G. The numbers \(n_{\min}\) and \(n_{\max}\) depend on the genus of \(\Pi\) /G, its orientability, the orders of the rotation centers of \(\Pi\) /G and the orders of the dihedral centers of \(\Pi\) /G. For each of the seventeen plane cristallographic groups the bounds \(n_{\min}\) and \(n_{\max}\) are explicitly calculated, and the combinatorially non-equivalent fundamental polygons are obtained.