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 \(\Gamma\) in \(\mathrm{Aut}(I(p,v))\) such that \(\Gamma\setminus I(p,v)\) 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 \(p\ge 6\), every uniform lattice in \(\mathrm{Aut}(I)\) contains a surface subgroup. We use elementary group theory, combinatorics, algebraic topology, and number theory.