A note on degenerations of Morse actions

We study Morse representations of discrete subgroups in higher rank semi-simple Lie groups defined by M. Kapovich, B. Leeb and J. Porti. We show that, if a sequence of Morse representations ${\rho }_n:\mathrm{\Gamma }\to G$ is (strongly) unbounded in the character variety, the group must have a very particular structure.