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 X-G/Gamma associated with arithmetic subgroups Gamma 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(infinity) of real points of the Q-group Res(k/Q)(G), obtained by restriction of scalars, is the finite direct product G(infinity) = Pi(v is an element of k,infinity) G(v) = SL2(R)(s) x SL2(C)(t), where the product ranges over the set V-k,V-infinity 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 X-G. Using reduction theory, one can construct an open subset Y-Gamma subset of X-G/Gamma such that its closure (Y) over bar (Gamma) is a compact manifold with boundary partial derivative(Y) over bar Gamma, and the inclusion (Y) over bar (Gamma) -> X-G/Gamma is a homotopy equivalence. The connected components Y-[P] of the boundary partial derivative(Y) over bar (Gamma) are in one-to-one correspondence with the finite set of G-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 T-m of dimension m = s + 2t = [k : Q], and its structure group is SLm(Z). Finally, we determine the cohomology of Y-[P].