In mathematics, a Bianchi group is a group of the form
P
S
L
2
(
O
d
)
where d is a positive square-free integer. Here, PSL denotes the projective special linear group and
O
d
is the ring of integers of the imaginary quadratic field
Q
(
−
d
)
.
The groups were first studied by Bianchi (1892) as a natural class of discrete subgroups of
P
S
L
2
(
C
)
, now termed Kleinian groups.
As a subgroup of
P
S
L
2
(
C
)
, a Bianchi group acts as orientation-preserving isometries of 3-dimensional hyperbolic space
H
3
. The quotient space
M
d
=
P
S
L
2
(
O
d
)
∖
H
3
is a non-compact, hyperbolic 3-fold with finite volume, which is also called Bianchi manifold. An exact formula for the volume, in terms of the Dedekind zeta function of the base field
Q
(
−
d
)
, was computed by Humbert as follows. Let
D
be the discriminant of
Q
(
−
d
)
, and
Γ
=
S
L
2
(
O
d
)
, the discontinuous action on
H
, then
v
o
l
(
Γ
∖
H
)
=
|
D
|
3
2
4
π
2
ζ
Q
(
−
d
)
(
2
)
.
The set of cusps of
M
d
is in bijection with the class group of
Q
(
−
d
)
. It is well known that any non-cocompact arithmetic Kleinian group is weakly commensurable with a Bianchi group.
