Abstract
Given a totally real algebraic number field k of degree s, we consider locally symmetric spaces
X
G
/
Γ
{X_{G}/\Gamma}
associated with arithmetic subgroups Γ of the special linear algebraic k-group
G
=
SL
M
2
(
D
)
{G=\mathrm{SL}_{M_{2}(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
Y
Γ
⊂
X
G
/
Γ
{Y_{\Gamma}\subset X_{G}/\Gamma}
such that its closure
Y
¯
Γ
{\overline{Y}_{\Gamma}}
is a compact manifold with boundary
∂
Y
¯
Γ
{\partial\overline{Y}_{\Gamma}}
, and the inclusion
Y
¯
Γ
→
X
G
/
Γ
{\overline{Y}_{\Gamma}\rightarrow X_{G}/\Gamma}
is a homotopy equivalence. The connected components
Y
[
P
]
{Y^{[P]}}
of the boundary
∂
Y
¯
Γ
{\partial\overline{Y}_{\Gamma}}
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
T
s
-
1
{T^{s-1}}
of dimension
s
-
1
{s-1}
, has the compact fibre
T
4
s
{T^{4s}}
, and its structure group is
SL
4
s
(
ℤ
)
{\mathrm{SL}_{4s}(\mathbb{Z})}
. We determine the cohomology of
Y
[
P
]
{Y^{[P]}}
. Secondly, if the quaternion division k-algebra D is indefinite, thus, there exists at least one archimedean place
v
∈
V
k
,
∞
{v\in V_{k,\infty}}
at which
D
v
{D_{v}}
splits over
ℝ
{\mathbb{R}}
, that is,
D
v
≅
M
2
(
ℝ
)
{D_{v}\cong M_{2}(\mathbb{R})}
, the fibre is homeomorphic to
T
4
s
{T^{4s}}
, but the base space of the bundle is more complicated.