A note on boundary components of arithmetic quotients of the group SL2 over an algebraic number field

Given an algebraic number field k, we consider quotients XG/Γ associated with arithmetic subgroups Γ of the special linear algebraic k-group G = SL2. The group G is k-simple, of k-rank one, and split over k. The Lie group G∞ of real points of the Q-group Resk/Q(G), obtained by restriction of scalars, is the finite direct product G∞ = ∏v∈Vk,∞ = SL2(R)s × SL2(C)t, where the product ranges over the set Vk,∞ of all archimedean places of k, and s (resp. t) denotes the number of real (resp. complex) places of k. The corresponding symmetric space is denoted by XG. Using reduction theory, one can construct an open subset YΓ ⊂ XG/Γ such that its closure YΓ is a compact manifold with boundary ∂YΓ, and the inclusion YΓ → XG/Γ is a homotopy equivalence. The connected components Y[P] of the boundary ∂YΓ are in one-to-one correspondence with the finite set of Γ-conjugacy classes of minimal parabolic k-subgroups of G. We are concerned with the geometric structure of the boundary components. Each component carries the natural structure of a fibre bundle. We prove that the basis of this bundle is homeomorphic to the torus Ts+t-1 of dimension s + t - 1, has the compact fibre Tm of dimension m = s + 2t = [k : Q], and its structure group is SLm(Z). Finally, we determine the cohomology of Y[P]