On some problems concerning discrete subgroups (Q2910128)

In the article under review four problems concerning discrete groups are considered. The first is a simple proof that there exists Fuchsian groups whose Selberg zeta-function has exceptional zeros. Or equivalently, that the corresponding Laplace operator has small eigenvalues. The proof is similar to \textit{B. Randol}'s [Bull. Am. Math. Soc. 80, 996--1000 (1974; Zbl 0317.30017)]. Next, the author gives examples of co-compact non-congruence subgroups of a compact arithmetic group, and non-congruence subgroups of the modular group. Next an explicit construction of modular forms are given for non-congruence subgroups of the modular group. Finally, a presentation is determined for the Hilbert modular group of \(\mathbb{Q}(\sqrt{5}).\)