In
mathematics, more precisely in
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
and
hyperbolic geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P ...
, Arithmetic Kleinian groups are a special class of
Kleinian group
In mathematics, a Kleinian group is a discrete subgroup of the group of orientation-preserving isometries of hyperbolic 3-space . The latter, identifiable with , is the quotient group of the 2 by 2 complex matrices of determinant 1 by their ...
s constructed using
orders in
quaternion algebra
In mathematics, a quaternion algebra over a field ''F'' is a central simple algebra ''A'' over ''F''See Milies & Sehgal, An introduction to group rings, exercise 17, chapter 2. that has dimension 4 over ''F''. Every quaternion algebra becomes a ma ...
s. They are particular instances of
arithmetic group
In mathematics, an arithmetic group is a group obtained as the integer points of an algebraic group, for example \mathrm_2(\Z). They arise naturally in the study of arithmetic properties of quadratic forms and other classical topics in number the ...
s. An arithmetic hyperbolic three-manifold is the quotient of
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. The ...
by an arithmetic Kleinian group.
Definition and examples
Quaternion algebras
A quaternion algebra over a field
is a four-dimensional
central simple -algebra. A quaternion algebra has a basis
where
and
.
A quaternion algebra is said to be split over
if it is isomorphic as an
-algebra to the algebra of matrices
; a quaternion algebra over an algebraically closed field is always split.
If
is an embedding of
into a field
we shall denote by
the algebra obtained by
extending scalars from
to
where we view
as a subfield of
via
.
Arithmetic Kleinian groups
A subgroup of
is said to be ''derived from a quaternion algebra'' if it can be obtained through the following construction. Let
be a
number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a f ...
which has exactly two embeddings into
whose image is not contained in
(one conjugate to the other). Let
be a quaternion algebra over
such that for any embedding
the algebra
is isomorphic to the
Hamilton quaternions. Next we need an order
in
. Let
be the group of elements in
of reduced norm 1 and let
be its image in
via
. We then consider the Kleinian group obtained as the image in
of
.
The main fact about these groups is that they are discrete subgroups and they have finite covolume for the
Haar measure on
. Moreover, the construction above yields a cocompact subgroup if and only if the algebra
is not split over
. The discreteness is a rather immediate consequence of the fact that
is only split at its complex embeddings. The finiteness of covolume is harder to prove.
An ''arithmetic Kleinian group'' is any subgroup of
which is
commensurable to a group derived from a quaternion algebra. It follows immediately from this definition that arithmetic Kleinian groups are discrete and of finite covolume (this means that they are
lattices in
).
Examples
Examples are provided by taking
to be an
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 ...
,
and
where
is the
ring of integers of
(for example
and