Relative SU(2, 1)-character varieties and decomposable complex hyperbolic triangle groups

In this work, we consider relative character varieties for representations of the 3-punctured sphere group in SU(2, 1). We provide necessary and sufficient conditions on the peripheral conjugacy classes, for such a representation to admit a decomposition as products of special elliptic elements. We apply these results to representations of the even subgroup of triangle groups, and describe components of the associated character variety.