Relatively geometric actions of Kähler groups on CAT(0) cube complexes

Abstract

We prove that for $n\ge 2$, 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 Kähler 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.

arXiv comment10 pages, 1 figure. This is a substantial enhancement of the previous version. In particular, Theorem 1.3 concerning relatively geometric actions of Kähler groups on CAT(0) cube complexes is completely new. Theorem 1.3 is then used to give a different proof of Theorem 1.1 than is found in the previous version

References in corpus (7)

• Peripheral fillings of relatively hyperbolic groups
• Groups acting on CAT(0) cube complexes
• Ubiquitous quasi-Fuchsian surfaces in cusped hyperbolic 3-manifolds
• Hyperbolic groups acting improperly
• Relative cubulations and groups with a 2-sphere boundary
• Cubulable Kähler groups
• Relatively geometric actions on CAT(0) cube complexes