In this paper we obtain presentations of fundamental groups of the complements of three quadric-line arrangements in P 2 . The first arrangement is a smooth quadric Q with n tangent lines to Q, and the second one is a quadric Q with n lines passing through a point p / ∈ Q. The last arrangement consists of a quadric Q with n lines passing through a point p ∈ Q.

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Some ball-quotient orbifolds are related by covering maps. We exploit these coverings to find infinitely many orbifolds on P 2 uniformized by the complex 2-ball B 2 and some orbifolds over K3 surfaces uniformized by B 2 . We also give, along with infinitely many reducible examples, an infinite series of irreducible curves along which P 2 is uniformized by the product of 1-balls B 1 × B 1 .

Abstract. We give an overview of the category of subgroups of the modular group, incorporating both the tame part, i.e. finite index subgroups, and the non-tame part, i.e. the rest. We also discuss arithmetic related questions which exist in both the tame part (via Belyi's theorem) and the non-tame part.