Surface quotients of right-angled hyperbolic buildings

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 .