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

TL;DR

This paper constructs examples of finitely generated infinite covolume subgroups of $PSL(2,R)^r$ with large limit sets, specifically those containing open subsets of the boundary, using semi-arithmetic Fuchsian groups with modular embeddings.

Contribution

It provides explicit examples of such subgroups with big limit sets, expanding understanding of their geometric boundary behavior.

Findings

Examples of subgroups with big limit sets are constructed.

Semi-arithmetic Fuchsian groups with modular embeddings are used.

These subgroups have limit sets containing open subsets of the boundary.

Abstract

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.