Surface Quotients of Right-Angled Hyperbolic Buildings

TL;DR

This paper explores the existence of surface quotients of Fuchsian hyperbolic buildings, establishing conditions under which these buildings can be quotiented to produce compact surfaces, advancing understanding in hyperbolic geometry and group actions.

Contribution

It develops the theory of surface quotients of Fuchsian buildings and provides conditions for the existence of lattices with compact surface quotients.

Findings

Existence of discrete subgroups with compact surface quotients in Fuchsian buildings

Necessary conditions related to building symmetries for such lattices

A method to choose fundamental group generators via closed geodesics

Abstract

Hyperbolic buildings are central objects in both hyperbolic geometry and geometric group theory, exhibiting a wide range of intriguing characteristics, especially with respect to group actions. In this paper, we develop the theory of surface quotients of Fuchsian buildings. For a large family of Fuchsian buildings, we prove the existence of a discrete subgroup $\Gamma$ of the automorphism group of the Fuchsian building whose quotient is a compact surface without boundary. We also provide some necessary conditions for the existence of such lattices, in terms of the symmetries of the building. The proof is based on another result of ours that generators of the fundamental group of a tessellated surface can be chose by closed geodesics in the 1-skeleton of the tessellation, which is of independent interest.