Rad HAZU, Matematičke znanosti, Vol. 28 (2024), 131-149.

A NOTE ON BOUNDARY COMPONENTS OF ARITHMETIC QUOTIENTS OF THE GROUP SL₂ OVER AN ALGEBRAIC NUMBER FIELD

Joachim Schwermer

Abstract.   Given an algebraic number field k, we consider quotients X_G/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 Res_k/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 V_k,∞ 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_Γ ⊂ X_G/Γ such that its closure Y_Γ is a compact manifold with boundary ∂Y_Γ, and the inclusion Y_Γ → X_G/Γ is a homotopy equivalence. The connected components Y^[P] of the boundary ∂Y_Γ 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 T^s+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 SL_m(Z). Finally, we determine the cohomology of Y^[P].

2020 Mathematics Subject Classification.   11F75, 57N65, 11R52.

Key words and phrases.   Cohomology of arithmetic groups, Hilbert modular varieties.

DOI: https://doi.org/10.21857/yrvgqt3529

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