Surface Quotients of Hyperbolic Buildings

Let Ip,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 Kv,v. We investigate and mostly determine the set of triples (p, v, g) for which there exists a uniform lattice Γ = Γp,v,g in Aut(Ip,v) such that Γ \Ip,v is a compact orientable surface of genus g. Surprisingly, for some p and g the existence of Γp,v,g depends upon the value of v. The remaining cases lead to open questions in tessellations of surfaces and in number theory. Our construction of Γp,v,g as the fundamental group of a simple complex of groups, together with a theorem of Haglund, implies that for p≥ 6 every uniform lattice in Aut(Ip,v) contains a surface subgroup. We use elementary group theory, combinatorics, algebraic topology, and number theory.