On arithmetic quotients of the group SL₂ over a quaternion division k-algebra

Given a totally real algebraic number field k of degree s , we consider locally symmetric spaces <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>X</m:mi> <m:mi>G</m:mi> </m:msub> <m:mo>/</m:mo> <m:mi mathvariant="normal">Γ</m:mi> </m:mrow> </m:math> {XG/Γ} associated with arithmetic subgroups Γ of the special linear algebraic k -group <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>G</m:mi> <m:mo>=</m:mo> <m:msub> <m:mi>SL</m:mi> <m:mrow> <m:msub> <m:mi>M</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>D</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:msub> </m:mrow> </m:math> {G=SL_{M₂(D)}} , attached to a quaternion division k -algebra D . The group G is k -simple, of k -rank one, and non-split over k . Using reduction theory, one can construct an open subset <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>Y</m:mi> <m:mi mathvariant="normal">Γ</m:mi> </m:msub> <m:mo>⊂</m:mo> <m:mrow> <m:msub> <m:mi>X</m:mi> <m:mi>G</m:mi> </m:msub> <m:mo>/</m:mo> <m:mi mathvariant="normal">Γ</m:mi> </m:mrow> </m:mrow> </m:math> {YΓ⊂ XG/Γ} such that its closure <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mover accent="true"> <m:mi>Y</m:mi> <m:mo>¯</m:mo> </m:mover> <m:mi mathvariant="normal">Γ</m:mi> </m:msub> </m:math> {ȲΓ} is a compact manifold with boundary <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>∂</m:mo> <m:mo>⁡</m:mo> <m:msub> <m:mover accent="true"> <m:mi>Y</m:mi> <m:mo>¯</m:mo> </m:mover> <m:mi mathvariant="normal">Γ</m:mi> </m:msub> </m:mrow> </m:math> {̄Γ} , and the inclusion <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mover accent="true"> <m:mi>Y</m:mi> <m:mo>¯</m:mo> </m:mover> <m:mi mathvariant="normal">Γ</m:mi> </m:msub> <m:mo>→</m:mo> <m:mrow> <m:msub> <m:mi>X</m:mi> <m:mi>G</m:mi> </m:msub> <m:mo>/</m:mo> <m:mi mathvariant="normal">Γ</m:mi> </m:mrow> </m:mrow> </m:math> {ȲΓ→ XG/Γ} is a homotopy equivalence. The connected components <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>Y</m:mi> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mi>P</m:mi> <m:mo stretchy="false">]</m:mo> </m:mrow> </m:msup> </m:math> {Y[P]} of the boundary <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>∂</m:mo> <m:mo>⁡</m:mo> <m:msub> <m:mover accent="true"> <m:mi>Y</m:mi> <m:mo>¯</m:mo> </m:mover> <m:mi mathvariant="normal">Γ</m:mi> </m:msub> </m:mrow> </m:math> {̄Γ} are in one-to-one correspondence with the finite set of Γ-conjugacy classes of minimal parabolic k -subgroups of G . We show that each boundary component carries the natural structure of a torus bundle. Firstly, if the quaternion division k -algebra D is totally definite, that is, D ramifies at all archimedean places of k , we prove that the basis of this bundle is homeomorphic to the torus <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>T</m:mi> <m:mrow> <m:mi>s</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> </m:math> {Tˢ⁻¹} of dimension <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>s</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {s-1} , has the compact fibre <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>T</m:mi> <m:mrow> <m:mn>4</m:mn> <m:mo>⁢</m:mo> <m:mi>s</m:mi> </m:mrow> </m:msup> </m:math> {T⁴ˢ} , and its structure group is <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>SL</m:mi> <m:mrow> <m:mn>4</m:mn> <m:mo>⁢</m:mo> <m:mi>s</m:mi> </m:mrow> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>ℤ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {SL₄ₛ(Z)} . We determine the cohomology of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>Y</m:mi> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mi>P</m:mi> <m:mo stretchy="false">]</m:mo> </m:mrow> </m:msup> </m:math> {Y[P]} . Secondly, if the quaternion division k -algebra D is indefinite, thus, there exists at least one archimedean place <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>v</m:mi> <m:mo>∈</m:mo> <m:msub> <m:mi>V</m:mi> <m:mrow> <m:mi>k</m:mi> <m:mo>,</m:mo> <m:mi mathvariant="normal">∞</m:mi> </m:mrow> </m:msub> </m:mrow> </m:math> {v∈ Vk,∞} at which <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>D</m:mi> <m:mi>v</m:mi> </m:msub> </m:math> {Dᵥ} splits over <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ℝ</m:mi> </m:math> {R} , that is, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>D</m:mi> <m:mi>v</m:mi> </m:msub> <m:mo>≅</m:mo> <m:mrow> <m:msub> <m:mi>M</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>ℝ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> {Dᵥ M₂(R)} , the fibre is homeomorphic to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>T</m:mi> <m:mrow> <m:mn>4</m:mn> <m:mo>⁢</m:mo> <m:mi>s</m:mi> </m:mrow> </m:msup> </m:math> {T⁴ˢ} , but the base space of the bundle is more complicated.