Conformal measure rigidity for representations via self-joinings

Abstract

Let $\hat{I}“$ be a Zariski dense discrete subgroup of a connected simple real algebraic group $G_1$. We discuss a rigidity problem for discrete faithful representations $\ddot{I}:\hat{I}“\to G_2$ and a surprising role played by higher rank conformal measures of the associated self-joining group. Our approach recovers rigidity theorems of Sullivan, Tukia and Yue, as well as applies to Anosov representations, including Hitchin representations. More precisely, for a given representation $\ddot{I}$ with a boundary map $f$ defined on the limit set $\hat{I}›$, we ask whether the extendability of $\ddot{I}$ to $G$ can be detected by the property that $f$ pushes forward some $\hat{I}“$-conformal measure class $[\hat{I}{½}_{\hat{I}}“]$ to a $\ddot{I}(\hat{I}“)$-conformal measure class $[\hat{I}{½}_{\ddot{I}(\hat{I}“)}]$. When $\hat{I}“$ is of divergence type in a rank one group or when $\ddot{I}$ arises from an Anosov representation, we give an affirmative answer by showing that if the self-joining $\hat{I}{“}_{\ddot{I}}=(\text{id}\times \ddot{I})(\hat{I}“)$ is Zariski dense in $G_1\times G_2$, then the push-forward measures $(\text{id}\times f{)}_{\ast }\hat{I}{½}_{\hat{I}}“$ and $(f^{-1}\times \text{id}{)}_{\ast }\hat{I}{½}_{\ddot{I}(\hat{I}“)}$, which are higher rank $\hat{I}{“}_{\ddot{I}}$-conformal measures, cannot be in the same measure class.

36 pages, 1 figure, To appear in Advances in Math