Surface quotients of hyperbolic buildings

TL;DR

This paper explores the existence of uniform lattices in hyperbolic buildings that produce compact orientable surfaces of a given genus, revealing dependence on parameters and implications for surface subgroups.

Contribution

It determines the conditions under which such lattices exist in Bourdon's hyperbolic buildings, linking geometric, algebraic, and number-theoretic aspects.

Findings

Existence of uniform lattices depends on parameter v.

For p>=6, all uniform lattices contain a surface subgroup.

Open questions remain in tessellations and number theory.

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 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.