Surface quotients of hyperbolic buildings

Abstract

Let I(p,v) be Bourdon's building, the unique simply-connected 2-complex such that all 2-cells are regular right-angled hyperbolic p-gons and the link at each vertex is the complete bipartite graph K(v,v). We investigate and mostly determine the set of triples (p,v,g) for which there exists a uniform lattice Γ in Aut(I(p,v)) such that Γ\I(p,v) is a compact orientable surface of genus g. Surprisingly, the existence of Γ depends upon the value of v. The remaining cases lead to open questions in tessellations of surfaces and in number theory. Our construction of Γ, together with a theorem of Haglund, implies that for p>=6, every uniform lattice in Aut(I) contains a surface subgroup. We use elementary group theory, combinatorics, algebraic topology, and number theory.