Stability of Lattices and the Partition of Arithmetic Quotients*

then there are a number of properties it possesses that play an important role in analysis on the quotient Γ\H, for example in the construction of Eisenstein series and the proof of the Selberg trace formula: • The region HY is invariant under the group N as well as the discrete subgroup Γ ∩ P ; • The quotient by N(Γ ∩ P ) is isomorphic to the subset [Y,∞); • for Y ≥ 1, the canonical projection from Γ ∩ P\H to Γ\H when restricted to the quotient Γ ∩ P\HY embeds it as a neighbourhood of the cusp at infinity; • the complement of the image of HY in Γ\H is compact.