On decompositions of discrete groups of motions of the plane

Abstract

For a discrete group G of motions of the Euclidean or Lobachevskii plane E there are two concepts of decomposition : geometric decomposition of the action and group-theoretical decomposition into a free product with an amalgamated subgroup. Both are often used in the study of these groups, for instance, in (1), (8), and (11). In (10) it is shown that almost always the existence of one type of decomposition implies that of the other. However, there this connection is not proved explicitly, but only in the process of determining what groups or what actions are decomposable. For the fundamental groups of closed surfaces it is shown in (2) by geometric- topological methods that any decomposition into a free product with an amalgamated cyclic subgroup is determined by a closed curve that decomposes the surface. A similar result is obtained in (3) by combinatorial group-theoretical methods. Here we consider decompositions of an arbitrary group G of motions of the plane E with compact fundamental domain and we prove that in the amalgamated subgroup there occur some elements. These correspond to closed curves on the surface E/G. If the amalgamated subgroup is cyclic or dihedral, then the decomposition has two factors, and the amalgamated subgroup either corresponds to a closed curve on E/G or is generated by two reflections, the axes of which have a common simple perpendicular (Theorems 6.2 and 6.5). The proof is based, as in (3), on the generalized Nielsen method of cancellation from (9). I take the opportunity to express my thanks to E.B. Vinberg for his help in preparing the final version of this article.