  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
{\Gamma} in Aut(I(p,v)) such that {\Gamma}\I(p,v) is a compact orientable
surface of genus g. Surprisingly, the existence of {\Gamma} depends upon the
value of v. The remaining cases lead to open questions in tessellations of
surfaces and in number theory. Our construction of {\Gamma}, 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.
