Existence, uniqueness and homogeneity of certain hyperbolic buildings (Q1433403)

In this very interesting paper, the author introduces a local structural condition for hyperbolic buildings with \(2\)-dimensional apartments. Concerning the buildings considered in the paper under review, their apartments are tessellations of the hyperbolic plane into regular \(k\)-gons, each vertex being contained in \(2m\) such \(k\)-gons. The building has rank \(k\) and if we omit in the diagram the edges with label \((\infty)\), then we have a cycle of length \(k\). The residues corresponding to the edges of these cycles are all generalized \(m\)-gons. The local structural condition assumed by the author is satisfied by a unique (locally finite!) building of the type above. Moreover, the building is homogeneous (has a transitive automorphism group). This is a very beautiful and deep result. Not many constructions or existence results of buildings take the automorphism group into account. The local conditions set by the author assume a preconceived finite classical generalized \(m\)-gon, and all rank 2 residues in the building are isomorphic either to that \(m\)-gon, or to a tree. As far as the reviewer can see, all finite classical polygons can be used here.