Bianchi group

In mathematics, a Bianchi group is a group of the form

PSL

2

(

O

d

)

{\displaystyle {\text{PSL}}_{2}({\mathcal {O}}_{d})}

where d is a positive square-free integer. Here,

PSL

2

{\displaystyle {\text{PSL}}_{2}}

denotes the projective special linear group and

O

d

{\displaystyle {\mathcal {O}}_{d}}

is the ring of integers of the imaginary quadratic field

Q

(

−

d

)

{\displaystyle \mathbb {Q} ({\sqrt {-d}})}

.

The groups were first studied by Bianchi (1892) as a natural class of discrete subgroups of

PSL

2

(

C

)

{\displaystyle {\text{PSL}}_{2}(\mathbb {C} )}

, now termed Kleinian groups.

Geometry

As a subgroup of

PSL

2

(

C

)

{\displaystyle {\text{PSL}}_{2}(\mathbb {C} )}

, a Bianchi group acts by orientation-preserving isometries on three-dimensional hyperbolic space

H

3

{\displaystyle \mathbb {H} ^{3}}

. The quotient

M

d

=

PSL

2

(

O

d

)

∖

H

3

{\displaystyle M_{d}={\text{PSL}}_{2}({\mathcal {O}}_{d})\backslash \mathbb {H} ^{3}}

is a non-compact, hyperbolic 3-fold of finite volume, called a Bianchi orbifold. An exact formula for its volume in terms of the Dedekind zeta function of the underlying imaginary quadratic field was computed by Humbert. If

D

{\displaystyle D}

is the discriminant of

Q

(

−

d

)

{\displaystyle \mathbb {Q} ({\sqrt {-d}})}

, then

vol

⁡

(

M

d

)

=

|

D

|

3

/

2

4

π

2

ζ

Q

(

−

d

)

(

2

)

.

{\displaystyle \operatorname {vol} (M_{d})={\frac {|D|^{3/2}}{4\pi ^{2}}}\zeta _{\mathbb {Q} ({\sqrt {-d}})}(2).}

The cusps of

M

d

{\displaystyle M_{d}}

are in bijection with the ideal classes of

Q

(

−

d

)

{\displaystyle \mathbb {Q} ({\sqrt {-d}})}

. Consequently, the number of cusps is equal to the class number of the field.

Arithmetic properties

Bianchi groups are arithmetic Kleinian groups.