Rigidity of convex co-compact diagonal actions

Abstract

Kleiner–Leeb and Quint showed that convex subsets in higher-rank symmetric spaces are very rigid compared to rank one symmetric spaces. Motivated by this, we consider convex subsets in products of proper $\mathrm{CAT}(0)$ spaces $X_1\times X_2$ and show that for any two convex co-compact actions ${\rho }_i(\Gamma )$ on $X_i$, where $i=1,2$, if the diagonal action of $\Gamma$ on $X_1\times X_2$ via $\rho =({\rho }_1,{\rho }_2)$ is also convex co-compact, then under a suitable condition, ${\rho }_1(\Gamma )$ and ${\rho }_2(\Gamma )$ have the same marked length spectrum.