Rigidity of convex co-compact diagonal actions

TL;DR

This paper investigates the rigidity properties of convex co-compact diagonal actions on products of CAT(0) spaces, showing under certain conditions that the actions share the same marked length spectrum, extending known rigidity results.

Contribution

It establishes a new rigidity result for convex co-compact actions on product spaces, linking the marked length spectra of component actions under diagonal convex co-compactness.

Findings

Convex subsets in higher-rank symmetric spaces exhibit strong rigidity.

Diagonal convex co-compact actions imply equal marked length spectra for component actions.

The result extends rigidity phenomena to products of CAT(0) spaces.

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 $\rho_i(\Gamma)$ 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.