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

Abstract

AbstractLet $$\Gamma $$ Γ be a discrete subgroup of $$\text {PU}(1,n)$$ PU ( 1 , n ) . In this work, we look at the induced action of $$\Gamma $$ Γ on the projective space $$\mathbb {P}(\wedge ^{k+1}\mathbb {C}^{n+1})$$ P ( ∧ k + 1 C n + 1 ) by the Plücker embedding, where $$\wedge ^{k+1}$$ ∧ 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 $$L(\Gamma )$$ L ( Γ ) , and we show several of its properties. We prove that its Kulkarni limit set is the union taken over all $$p\in L(\Gamma )$$ p ∈ L ( Γ ) of the projective subspace generated by all k-planes that contain p or are contained in $$p^{\perp }$$ p ⊥ via the Plücker embedding. We also prove a duality between both limit sets.