In mathematics, an arithmetic hyperbolic 3-manifold is a hyperbolic 3-manifold

Hyperbolic 3-manifold

A hyperbolic 3-manifold is a 3-manifold equipped with a complete Riemannian metric of constant sectional curvature -1. In other words, it is the quotient of three-dimensional hyperbolic space by a subgroup of hyperbolic isometries acting freely and properly discontinuously...

 whose fundamental group

Fundamental group

In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...

 is an arithmetic group

Arithmetic group

In mathematics, an arithmetic group in a linear algebraic group G defined over a number field K is a subgroup Γ of G that is commensurable with G, where O is the ring of integers of K. Here two subgroups A and B of a group are commensurable when their intersection has finite index in each of them...

 as a subgroup of PGL(2,C). The one of smallest volume is the Weeks manifold

Weeks manifold

In mathematics, the Weeks manifold, sometimes called the Fomenko–Matveev–Weeks manifold, is a closed hyperbolic 3-manifold obtained by and Dehn surgeries on the Whitehead link. It has volume approximately equal to 0.9427... and showed that it has the smallest volume of any closed orientable...

, and the one of next smallest volume is the Meyerhoff manifold

Meyerhoff manifold

In hyperbolic geometry, the Meyerhoff manifold is the arithmetic hyperbolic 3-manifold obtained by surgery on the figure-8 knot complement. It was introduced by as a possible candidate for the hyperbolic 3-manifold of smallest volume, but the Weeks manifold turned out to have slightly smaller...

.

Trace fields

The trace field of a Kleinian group

Kleinian group

In mathematics, a Kleinian group is a discrete subgroup of PSL. The group PSL of 2 by 2 complex matrices of determinant 1 modulo its center has several natural representations: as conformal transformations of the Riemann sphere, and as orientation-preserving isometries of 3-dimensional hyperbolic...

 Γ is the field generated by the traces of representatives of its elements in SL(2, C) and it is denoted by tr Γ. The trace field of a finite covolume Kleinian group is an algebraic number field

Algebraic number field

In mathematics, an algebraic number field F is a finite field extension of the field of rational numbers Q...

, a finite extension of the rational number

Rational number

In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...

s.

The invariant trace field of a Kleinian group Γ is the trace field of the Kleinian group Γ⁽²⁾ generated by squares of elements of Γ.

The quaternion algebra of a Kleinian group Γ is the subring

Subring

In mathematics, a subring of R is a subset of a ring, is itself a ring with the restrictions of the binary operations of addition and multiplication of R, and which contains the multiplicative identity of R...

of M(2, C) generated by the trace field and the elements of Γ, and is a 4-dimensional simple algebra over the trace field if Γ is not elementary. The invariant quaternion algebra of Γ is the quaternion algebra of Γ⁽²⁾. The quaternion algebra may be split, in other words a matrix algebra; this happens whenever Γ is non-elementary and has a parabolic element, in particular if it is a Kleinian group of non-compact finite covolume 3-manifold.