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 1 symmetric spaces. Motivated by this, we consider convex subsets in products of proper CAT(0) spaces $X_1\times X_2$ and show that for any two convex co-compact actions $\ddot{I}{}_i(\hat{I}“)$ on $X_i$, where $i=1,2$, if the diagonal action of $\hat{I}“$ on $X_1\times X_2$ via $\ddot{I}=(\ddot{I}{}_1,\ddot{I}{}_2)$ is also convex co-compact, then under a suitable condition, $\ddot{I}{}_1(\hat{I}“)$ and $\ddot{I}{}_2(\hat{I}“)$ have the same marked length spectrum.

10 pages