Embedding theorems for non-positively curved polygons of finite groups
A polygon of groups has groups associated to its vertices, its edges, and to its face. There is an inclusion of the face group into each edge group and an inclusion of each edge group into its associated vertex groups. It is assumed that at each vertex the two edge groups intersect only in the face group. The fundamental group of the polygon of groups is defined to be the free product of the vertex groups amalgamated along the edge groups. A concept of angle between the two edge groups at a vertex has been defined by Gersten and Stallings. If the resulting angles belong to a Euclidean or hyperbolic polygon, then the polygon of groups is called nonpositively curved. Gersten and Stallings proved that for a nonpositively curved polygon of groups, the vertex groups embed in the amalgamated free product. K. S. Brown proved that Thompson's infinite simple group is the fundamental group of a polygon of groups, and hence such groups need not be residually finite. That example, however, has positive curvature. In the paper under review, the authors show that there is a nonpositively curved triangle of finite groups whose fundamental group is not residually finite. To construct their examples, they prove two imbedding theorems that are of independent interest. The first states that if \(X\) is a complete squared complex, i. e. a 2-complex whose universal cover is isomorphic to the product of two trees, then for each prime \(p\) there is a right-angled square of finite \(p\)-groups \(R_p(X)\) such that \(\pi_1(X)\) imbeds into \(\pi_1(R_p(X))\). The second states that if \(R\) is a square of finite groups with all vertex angles \(\leq{\pi\over 2}\), then \(\pi_1(R)\) can be imbedded in the fundamental group of a nonpositively curved triangle of finite groups. Since there exist complete squared complexes with nonresidually finite fundamental groups, applying the two imbedding theorems yields the desired examples.
scientific article; zbMATH DE number 1121667