The Limit Set for Representations of Discrete Subgroups of $$\text {PU}(1,n)$$ by the Plücker Embedding

Abstract

Let \(\Gamma \) be a discrete subgroup of \(\text {PU}(1,n)\). In this work, we look at the induced action of \(\Gamma \) on the projective space \(\mathbb {P}(\wedge ^{k+1}\mathbb {C}^{n+1})\) by the Plücker embedding, where \(\wedge ^{k+1}\) denotes the exterior power. We define a limit set for this action called the k-Chen-Greenberg limit set, which extends the classical definition of the Chen-Greenberg limit set \(L(\Gamma )\), and we show several of its properties. We prove that its Kulkarni limit set is the union taken over all \(p\in L(\Gamma )\) of the projective subspace generated by all k-planes that contain p or are contained in \(p^{\perp }\) via the Plücker embedding. We also prove a duality between both limit sets.