Recent work of Alessandrini-Lee-Schaffhauser generalized the theory of higher Teichmüller spaces to the setting of orbifold surfaces. In particular, these authors proved that, as in the torsion-free surface case, there is a "Hitchin component" of representations into PGL(n,R) which is homeomorphic to a ball. They explicitly compute the dimension of Hitchin components for triangle groups, and find that this dimension is positive except for a finite number of low-dimensional examples where the representations are rigid. In contrast with these results and with the torsion-free surface group case, we show that the composition of the geometric representation of a hyperbolic triangle group with a diagonal embedding into PGL(2n,R) or PSp(2n,R) is always locally rigid.
Fanny Kassel