Trivalent Polygonal Complexes of Nonpositive Curvature and Platonic Symmetry

Abstract

In this paper we study a natural generalization of Platonic solids: two-dimensional simply connected polygonal complexes with flag transitive group of combinatorial automorphisms. Our results give an almost complete description of such symmetric complexes with constant valency 3. The initial local data for the construction of such a complex are a regular k-gon and a (highly symmetric) graph L: the link at a vertex. We assume nonpositive curvature for the complex. This greatly simplifies the question of existence and the real issue is the uniqueness. The main ingredient of our analysis is the theory of regular graphs, a well-developed subject with 50 years of history. Delicate symmetry properties of these graphs yield a variety of local phenomena in complexes and provide the appropriate tool to study the uniqueness question. We should point out that many examples of the complexes we consider have already appeared in the literature, most prominently two-dimensional Bruhat—Tits buildings, but there have also been more recent constructions. We show that many of the symmetric complexes have nondiscrete automorphism groups. Clearly buildings are in this class, and some other examples were previously constructed. These automorphism groups resemble p-adic Lie groups and their further study should be worthwhile.