Equicontinuity Regions and Kulkarni Limit Sets for Discrete Subgroups of Quaternionic Hyperbolic Isometry Groups Acting on Grassmannians

Let $\Gamma$ be a discrete subgroup of $\mathrm{PSp}(n,1)$. We realize $\mathbb{H}^{n+1}$ as $\mathbb{C}^{2n+2}$ and study the induced action of $\Gamma$ on $\mathbb P (\wedge_\mathbb{C}^{2k+2} \mathbb{C}^{2n+2})$ where $0\le k\le n-1$, into which the quaternionic Grassmannian $\mathrm{Gr}_{k+1}(\mathbb H^{n+1})$ embeds equivariantly. We determine all pseudo-projective limits of the action. Their images are the linear spans of the $(k+1)$-planes tangent to the boundary at points of the Chen--Greenberg limit set, and their kernels are the orthogonal complements of such spans. As a consequence, we describe the equicontinuity region explicitly, identify the $k$-Chen-Greenberg limit set with the set of tangent $(k+1)$-planes at limit points, and show that the equicontinuity region is contained in the Kulkarni domain of discontinuity. For non-elementary groups, we also give a lower bound for the Kulkarni limit set in terms of the $k$-Chen-Greenberg limit set.