Surface quotients of right-angled hyperbolic buildings

Abstract

In this paper, we develop the theory of surface quotients of Fuchsian buildings, a hyperbolic building in which each chamber is a regular $p$-gon. Our focus is on the right-angled Fuchsian building $I_{p,\mathbf{q}}$, defined by a $p$-tuple $\mathbf{q}$ of integers, which indicates the thickness of each side of the chamber. We proved some sufficient conditions for the existence of a surface quotient for a given compact hyperbolic surface tessellated by right-angled regular $p$-gons. By combining these results with tessellation methods of hyperbolic surfaces, we find some conditions on $\mathbf{q}$ under which a surface quotient of $I_{p,\mathbf{q}}$ exists. We also define certain symmetry conditions of $\mathbf{q}$ and prove that these symmetries are necessary for the existence of a surface quotient of $I_{p,\mathbf{q}}$ when the number of $p$-gons to tessellate the surface quotient is not divisible by $4$.