Exceptional points for finitely generated Fuchsian groups of the first kind (Q2663683)

Let \(G\) be a finitely generated Fuchsian group of the first kind and let \((g:m_1,\dots, m_r;s)\) be its signature. In the paper under review, this signature is shortened to \((g:m_1, \dots, m_n)\) considering parabolic elements of \(G\) as elliptic elements of infinite order. \textit{A. F. Beardon} in [Graduate Texts in Mathematics, 91, Springer-Verlag XII (1983; Zbl 0528.30001)] showed that almost every Dirichlet region associated to the action of such a group has \(12g + 4n - 6\) sides. Points in the hyperbolic plane \(\mathbb{H}\) which admit a Dirichlet region with fewer sides are called exceptional points. The remaining points are called regular points. In the main result of the paper under review, Theorem 4.2, it is proved that if \(G\) is a Fuchsian group satisfying the above conditions then there exist uncountable many exceptional points for \(G\) in \(\mathbb{H}\). In Section 3, several topological results about regular points are obtained. In particular, it is proved (Theorem 3.5) that the set of all regular points for \(G\) is an open subset of \(\mathbb{H}\). Finally, Theorem 4.2 is proved in Section 4. The authors point out that a mention to exceptional points appears in only two other papers, but in both papers the considered Fuchsian groups are cocompact.