Horn's problem in PU$(n,1)$

The multiplicative Horn problem is the following question: given three conjugacy classes ${\mathcal{C}}_1,{\mathcal{C}}_2,{\mathcal{C}}_3$ in a Lie group $G$, do there exist elements $(A,B,C)\in {\mathcal{C}}_1\times {\mathcal{C}}_2\times {\mathcal{C}}_3$ such that $ABC=\mathrm{Id}$? In this paper, we study the multiplicative Horn problem restricted to the elliptic classes of the group $G=\mathrm{P}\mathrm{U}(n,1)$ for $n\ge 1$, which is the isometry group of the $n$-dimensional complex hyperbolic space. We show that the solution set of Horn’s problem in PU$(n,1)$ is a finite union of convex polytopes in the space of elliptic conjugacy classes. We give a complete description of these polytopes when $n=2$.