Approximating hyperbolic lattices by cubulations

Abstract

We show that an isometric action of a torsion-free uniform lattice $\Gamma$ on hyperbolic space $\mathbb{H}^n$ can be metrically approximated by geometric actions of $\Gamma$ on $\mathrm{CAT}(0)$ cube complexes, provided that either $n$ is at most three, or the lattice is arithmetic of simplest type. This solves a conjecture of Futer and Wise. Our main tool is the study of a space of co-geodesic currents, consisting of invariant Radon measures supported on codimension-1 hyperspheres in the Gromov boundary of $\mathbb{H}^n$. By pairing co-geodesic currents and geodesic currents via an intersection number, we show that asymptotic convergence of geometric actions can be deduced from the convergence of their dual co-geodesic currents. For surface groups, our methods also imply approximation by cubulations for actions induced by non-positively curved Riemannian surfaces with singularities, Hitchin and maximal representations, and quasiFuchsian representations.