Examples of infinite covolume subgroups of $PSL(2,R{)}^r$ with big limit sets

We provide examples of finitely generated infinite covolume subgroups of $PSL(2,R{)}^r$ with a “big” limit set, e.g. that contains an open subset of the geometric boundary. They are given by the so called semi-arithmetic Fuchsian groups admitting modular embeddings.