Surface quotients of hyperbolic buildings

David Futer and Anne Thomas

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 $\mathrm{\Gamma }$ in $\mathrm{Aut}(I(p,v))$ such that $\mathrm{\Gamma }\setminus I(p,v)$ is a compact orientable surface of genus $g$. Surprisingly, the existence of $\mathrm{\Gamma }$ depends upon the value of $v$. The remaining cases lead to open questions in tessellations of surfaces and in number theory. Our construction of $\mathrm{\Gamma }$, together with a theorem of Haglund, implies that for $p\ge 6$, every uniform lattice in $\mathrm{Aut}(I)$ contains a surface subgroup. We use elementary group theory, combinatorics, algebraic topology, and number theory.