The growth of nonpositively curved triangles of groups (Q1365482)

If \(E\) and \(F\) are subgroups of \(V\), and \(X\to E\) and \(X\to F\) are group monomorphisms, then there is a natural homomorphism from the free product with amalgamation \(E*_XF\) to \(V\). The shortest nontrivial word in its kernel has length \(2n\) for some \(n\), and the angle between \(E\) and \(F\) (with respect to \(X\) and the monomorphisms) is defined to be \(\pi/n\), or \(0\) if the homomorphism is injective. A triangle \(T\) of groups is a collection of seven groups corresponding to the face, the three edges, and the three vertices of \(T\), together with monomorphisms from the face group into each edge group, and from each edge group into the groups of the two vertices it contains. To each vertex is associated the angle between its two edge subgroups, and \(T\) is called nonpositively curved if the sum of the angles is no more than \(\pi\). A triangle of groups has a ``pushout'' group \(G_T\), and J. Stallings showed that if \(T\) is nonpositively curved, then the homomorphism from each vertex group into \(G_T\) is injective. Associated to \(T\) is a 2-complex \(X_T\) called the amalgam complex, which admits a simplicial action of \(G_T\). Its vertices correspond to the left cosets of the images of the vertex groups in \(G_T\), its edges to the left cosets of the images of the edge groups, and its faces to the left cosets of the image of the face group. Assume now that \(T\) is nonpositively curved and that its face group is trivial. One can write \(X_T\) as an increasing union of combinatorial balls \(X(n)\), where \(X(0)\) is a single 2-simplex and inductively \(X(n+1)\) is the union of the simplices that meet \(X(n)\). The main technical result of the paper details the local structure of the \(X(n)\) and their boundaries \(\partial X(n)\). In the ensuing applications, it is assumed that the vertex groups are finite. First, a regular combing for \(G_T\) is developed using the combinatorics of \(X_T\); in particular, \(G_T\) is automatic. Then, growth functions \(f\) of \(G_T\) are studied. Roughly speaking, certain local types of vertices are defined, and the authors prove that there is a linear recursion which determines the numbers of vertices of each type in \(\partial X(n+1)\) in terms of the numbers of each type in \(\partial X(n)\). This leads to an effective description of \(f\). If no vertex angle of \(T\) is \(\pi/2\), there are at most 9 vertex types, and \(f\) is a rational function of degree at most 6 whose coefficients are polynomials in the orders of the edge groups and vertex groups of \(G\). If an angle of \(\pi/2\) is allowed, there is no bound on the number of vertex types of \(T\) as \(T\) varies. Nonetheless, the authors are able to prove that the degree of \(f\) is always at most 10, and a symbolic computation of their expression for \(f\) shows that the degree is at most 7. In fact, vertex angles of \(\pi/2\) cause considerable complication throughout the paper. Numerous examples of triangles of groups are provided. These include two triangles for which each edge group has order 3 and each vertex group is the nonabelian group of order 39. Both triangles have angle \(\pi/3\) at each vertex, yet one \(G_T\) is negatively curved and the other is not (since it contains a \(\mathbb{Z}\times\mathbb{Z}\) subgroup). Other examples include a triangle with vertex angles \(\pi/2\), \(\pi/4\), and \(\pi/4\) and \(G_T\) isomorphic to \(\text{PGL}(2,\mathbb{Z}[i])\), and triangles with \(G_T\) the fundamental groups of tetrahedral orbifolds with a vertex at infinity. The authors explain that allowing the face group of \(T\) to be nontrivial adds complications which ``though minor \(\ldots\) seem to outweigh the completeness achieved by considering general face groups carefully. The results of this paper hold for general face groups with at most minor modifications''.