Discrete Subgroups of $\text{ PSL }(n+1,{\mathbb {C}})$ Acting on the Grassmannians

Abstract

Let $\varGamma$ be a discrete subgroup of $\text{ PU(1, }n\text{) }$. In this work we look at the induced action of $\varGamma$ on the Grassmannian $\text{ Gr}_{k}(\mathbb {CP}^{n})$ of all k-planes in $\mathbb {CP}^{n}$ for each $0\le k <n$. We prove various analytic, geometric and dynamical properties of this action. In particular, we show that the equicontinuity region $\text{ Eq}^{k,n}(\varGamma )$ is the complement of the union of all k-planes that either pass through a point p in the Chen–Greenberg limit set $L(\varGamma )$ or are contained in the orthogonal hyperplane $p^{\perp }$, generalizing a known theorem for the case $k=0$. We also prove that the action of $\varGamma$ on $\text{ Eq}^{k,n}(\varGamma )$ is properly discontinuous.