Ergodic theory and rigidity on the symmetric space of non-compact type (Q2709592)

The author considers a subgroup \(\Gamma\) of the isometry group of a symmetric space \(X\) of noncompact type. He shows that the marked length spectrum function on \(\Gamma\) determines \(X\) and the conjugacy class of \(\Gamma\) under suitable assumptions on \(\Gamma\) and within a family of suitably chosen comparison groups.NEWLINENEWLINENEWLINEAs an example, it is shown that two Zariski dense subgroups of isometry groups of rank one symmetric spaces which admit a marked-length spectrum preserving isomorphism are conjugate subgroups of the same simple rank one Lie group.NEWLINENEWLINENEWLINEThe proofs rely on a detailed analysis of the cross ratio. The main point is the existence of an equivariant homeomorphism between the limit sets of two nonelementary and non-parabolic groups of isometries acting on CAT(-1)-spaces which are isomorphic with a marked-length spectrum preserving isomorphism.