In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a Bianchi group is a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic iden ...
of the form
:
where ''d'' is a positive
square-free integer
In mathematics, a square-free integer (or squarefree integer) is an integer which is divisible by no square number other than 1. That is, its prime factorization has exactly one factor for each prime that appears in it. For example, is square-f ...
. Here, PSL denotes the
projective special linear group
In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space ''V'' on the associate ...
and
is the ring of integers of the
imaginary quadratic field
In algebraic number theory, a quadratic field is an algebraic number field of degree two over \mathbf, the rational numbers.
Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free integer different from 0 an ...
.
The groups were first studied by as a natural class of
discrete subgroups of
, now termed
Kleinian group
In mathematics, a Kleinian group is a discrete subgroup of the group (mathematics), group of orientation-preserving Isometry, isometries of hyperbolic 3-space . The latter, identifiable with PSL(2,C), , is the quotient group of the 2 by 2 complex ...
s.
As a subgroup of
, a Bianchi group acts as
orientation-preserving
The orientation of a real vector space or simply orientation of a vector space is the arbitrary choice of which ordered bases are "positively" oriented and which are "negatively" oriented. In the three-dimensional Euclidean space, right-handed ...
isometries
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...
of 3-dimensional
hyperbolic space
In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to -1. It is homogeneous, and satisfies the stronger property of being a symmetric space. Th ...
. The quotient space
is a non-compact, hyperbolic 3-fold with finite volume, which is also called ''Bianchi orbifold''. An exact formula for the volume, in terms of the
Dedekind zeta function
In mathematics, the Dedekind zeta function of an algebraic number field ''K'', generally denoted ζ''K''(''s''), is a generalization of the Riemann zeta function (which is obtained in the case where ''K'' is the field of rational numbers Q). It ca ...
of the base field
, was computed by
Humbert
Humbert, Umbert or Humberto (Latinized ''Humbertus'') is a Germanic given name, from ''hun'' "warrior" and ''beraht'' "bright". It also came into use as a surname.
Given name
;Royalty and Middle Ages
* Emebert (died 710)
* Humbert of Maroilles ...
as follows. Let
be the discriminant of
, and
, the discontinuous action on
, then
:
The set of cusps of
is in bijection with the class group of
. It is well known that every non-cocompact arithmetic Kleinian group is weakly commensurable with a Bianchi group.
[Maclachlan & Reid (2003) p. 58]
References
*
*
*
*
*
External links
* Allen Hatcher
''Bianchi Orbifolds''
Group theory
{{Abstract-algebra-stub