Rigidity on symmetric spaces

AbstractIn this note we show the marked length rigidity of symmetric spaces. More precisely, if X and Y are symmetric spaces of noncompact type without Euclidean de Rham factor, with G1 and G2 corresponding real semisimple Lie groups, and Γ1⊂G1,Γ2⊂G2 are Zariski dense subgroups with the same marked length spectrum, then X=Y and Γ1,Γ2 are conjugate by an isometry. As an application, we answer in the affirmative a Margulis's question and show that the cross-ratio on the limit set determines the Zariski dense subgroups up to conjugacy. We also embed the space of nonparabolic representations from Γ to G into RΓ