Revisiting the Siegel upper half plane I

Abstract

In the first part of the paper we show that the Busemann 1-compactification of the Siegel upper half plane of rank n: SH n= Sp (n, R)/ K n 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 SH n . We show that the the set of accumulation points of the orbit Γ( Z) on the Shilov boundary of SH n 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).