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.

10 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 (1)

• Relatively geometric actions on CAT(0) cube complexes