Approximating hyperbolic lattices by cubulations
Approximating hyperbolic lattices by cubulations
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 …