Exceptional points for cocompact Fuchsian groups

Abstract

Let G be a cocompact Fuchsian group acting on the hyperbolic plane H. If G covers a compact hyperbolic surface of genus g ≥ 2, then almost every Dirichlet region for G has 12g −6 sides. In this article, we study the exceptional points for G, i.e., the points in H associated to Dirichlet regions for G with strictly less than 12g − 6 sides. More specifically, we show that uncountably many exceptional points exist for any cocompact group. We also define and prove the existence of higher order exceptional points for any such group.