Rank One Lattices Whose Parabolic Isometries Have No Rotational Part

We prove a result on certain finite index subgroups of rank one lattices which is motivated by cusp closing constructions. Let X always denote a rank one symmetric space of non-compact type, i.e., X is the hyperbolic space KH, n ≥ 2, where K is either R,C,H or O and n = 2 in the latter case. By Σ we always denote a lattice in the isometry group Iso(X), that is, a discrete subgroup of the isometry group Iso(X) of X such that the quotient Σ\X has finite volume. The result below is only relevant for non-uniform lattices, i.e. lattices Σ where Σ\X is non-compact. Such lattices exist for each X by a result of A. Borel (see [B1] or Chapter XIV in [R]). We say that a parabolic isometry σ ∈ Iso(X) has no rotational part if σ is contained in the nilpotent part N of some Iwasawa decomposition Iso(X) = NAK. Here Iso(X) denotes the identity component of Iso(X). In this note we give a simple geometric argument in order to prove the theorem below, which is related to a result of Borel and Garland & Raghunathan. Theorem. For any lattice Σ < Iso(X) in the isometry group of a rank one symmetric space X of non-compact type there exists a finite subset F ⊂ Σ of parabolic isometries such that the following holds. Assume Σ′ C Σ is a normal subgroup and Σ′ ∩ F = ∅. Then any parabolic isometry in Σ′ has no rotational part. Remark. The proof of the theorem provides an explicit procedure to determine F , which is in some sense optimal by the example at the end. As a consequence of the theorem we obtain the following corollary. The statement is proved in [GR], Lemma 6.5, with Proposition 17.6 from [B2], and the methods are of algebraic nature. Corollary (Borel and Garland & Raghunathan). Let V = Σ\X be a complete, locally rank one symmetric manifold of non-compact type and finite volume, and denote by N the nilpotent part of an Iwasawa decomposition of Iso(X). Then there exists a finite regular covering V̂ → V such that each cusp of V̂ is diffeomorphic to Γ\N × [0,∞) for some lattice Γ < N . Received by the editors December 7, 1996 and, in revised form, January 22, 1997. 1991 Mathematics Subject Classification. Primary 53C35; Secondary 22E40, 22E25.