Relatively geometric actions of K\"ahler groups on CAT(0) cube complexes

TL;DR

The paper proves limitations on relatively geometric actions of certain lattices and Kähler groups on CAT(0) cube complexes, revealing structural constraints and classifying specific group actions.

Contribution

It establishes non-existence results for non-uniform lattices in PU(n,1) acting relatively geometrically on CAT(0) cube complexes and classifies hyperbolic Kähler groups with such actions.

Findings

Non-uniform lattices in PU(n,1) do not admit relatively geometric actions on CAT(0) cube complexes for n≥2.

A non-uniform lattice in a semisimple Lie group without compact factors must be commensurable with SO(n,1) if it admits such an action.

Hyperbolic Kähler groups relatively hyperbolic to residually finite parabolic subgroups are virtually surface groups.

Abstract

We prove that for $n\geq 2$, a non-uniform lattice in $\text{PU}(n,1)$ does not admit a relatively geometric action on a $\mathrm{CAT}(0)$ cube complex, in the sense of Einstein and Groves. As a consequence, if $\Gamma$ is a non-uniform lattice in a non-compact semisimple Lie group $G$ without compact factors that admits a relatively geometric action on a $\mathrm{CAT}(0)$ cube complex, then $G$ is commensurable with $\text{SO}(n,1)$. We also prove that if a K\"ahler group is hyperbolic relative to residually finite parabolic subgroups, and acts relatively geometrically on a $\mathrm{CAT}(0)$ cube complex, then it is virtually a surface group.