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After Voronin proved the universality theorem of the Riemann zeta function inthe 1970s, universality theorems have been proposed for various zeta andL-functions. Drungilas-Garunkstis-Kacenas' work at 2013 on the universalitytheorem of the Selberg zeta function for the modular group is one of them andis probably the first universality theorem of the zeta function of ordergreater than one. Recently, Mishou (2021) extended it by proving the jointuniversality theorem for the principal congruence subgroups. In the presentpaper, we further extend these works by proving the (joint) universalitytheorem for subgroups of the modular group and co-compact arithmetic groupsderived from indefinite quaternion algebras, which is available in the regionwider than the regions in the previous two works.

Universality theorems of the Selberg zeta functions for arithmetic groups