Revisiting the Siegel Upper Half Plane I

In the first part of the paper we study -metrics on = (, ℂ)/(, ℂ) for ∈ [1, ∞]. We give a complete description of -Busemann compactification of for ∈ [1, ∞). For the Siegel upper half plane of rank = (, ℝ)/ we show that the 1-Busemann compactification is the compactification of as the bounded domain. In the second part of the paper we study certain properties of a discrete group Γ of biholomorphisms of . We show that the the set of accumulation points of the orbit Γ() on the Shilov boundary of is independent of , and denote this set by Λ(Γ). We associate with Γ the standard class of p-Patterson-Sullivan measures. For - regular Γ these measures are supported on Λ(Γ). For 1-regular Γ 1-Patterson-Sullivan measures are conformal densities. For Γ, with , we give a modified version of the class of Patterson-Sullivan measures, which are always supported on Λ(Γ)