Fundamental domains for lattices in rank one semisimple Lie groups

Abstract

We construct a fundamental domain Ω for an arbitrary lattice Γ in a real rank one, real simple Lie group, where Ω has finitely many cusps (i.e., is a finite union of Siegel sets) and has the Siegel property (i.e., the set {γ ∈ Γ|Ω γ ∩ Ω ≠ Φ} is finite). From the existence of Ω we derive a number of consequences. In particular, we show that Γ is finitely presentable and is almost always rigid.