Revisiting the Siegel upper half plane I

In the first part of the paper we show that the Busemann 1-compactification of the Siegel upper half plane of rank is the compactification as a bounded domain. In the second part of the paper we study certain properties of discrete groups Γ of biholomorphisms of SHn. We show that the the set of accumulation points of the orbit Γ(Z) on the Shilov boundary of SHn is independent of Z, and denote this set by Λ(Γ). We associate with Γ the standard class of Patterson–Sullivan (PS) p-measures. For p-regular Γ these measures are supported on Λ(Γ). For 1-regular Γ PS 1-measures are conformal densities. For Γ, with Λ(Γ)≠ ︀, we give a modified version of the class of PS measures, which are always supported on Λ(Γ).