Exceptional Points for Finitely Generated Fuchsian Groups of the First Kind
Let G be a finitely generated Fuchsian group of the first kind and let (g : m1, m2, …, mn) be its shortened signature. Beardon showed that almost every Dirichlet region for G has 12g + 4n − 6 sides. Points in ℍ corresponding to Dirichlet regions for G with fewer sides are called exceptional for G. We generalize previously established methods to show that, for any such G, its set of exceptional points is uncountable.
Fera, J, & Lazowski, A. (2019). Exceptional points for finitely generated Fuchsian groups of the first kind. In T. Grundhofer & M. Joswig (Eds.). Advances in Geometry. Berlin: Walter de Gruyter GmbH. doi: 10.1515/advgeom-2019-0013