In mathematics, more precisely in group theory and hyperbolic geometry, Arithmetic Kleinian groups are a special class of Kleinian groups constructed using orders in quaternion algebras.
the algebra obtained by extending scalars from
is said to be derived from a quaternion algebra if it can be obtained through the following construction.
We then consider the Kleinian group obtained as the image in
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
The discreteness is a rather immediate consequence of the fact that
The finiteness of covolume is harder to prove.
[1] 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
is any quaternion algebra over an imaginary quadratic number field
which is not isomorphic to a matrix algebra then the unit groups of orders in
The invariant trace field of a Kleinian group (or, through the monodromy image of the fundamental group, of an hyperbolic manifold) is the field generated by the traces of the squares of its elements.
In the case of an arithmetic manifold whose fundamental groups is commensurable with that of a manifold derived from a quaternion algebra over a number field
the invariant trace field equals
One can in fact characterise arithmetic manifolds through the traces of the elements of their fundamental group.
A Kleinian group is an arithmetic group if and only if the following three conditions are realised: For the volume of an arithmetic three manifold
derived from a maximal order in a quaternion algebra
A consequence of the volume formula in the previous paragraph is that This is in contrast with the fact that hyperbolic Dehn surgery can be used to produce infinitely many non-isometric hyperbolic 3-manifolds with bounded volume.
In particular, a corollary is that given a cusped hyperbolic manifold, at most finitely many Dehn surgeries on it can yield an arithmetic hyperbolic manifold.
The Weeks manifold is the hyperbolic three-manifold of smallest volume[3] and the Meyerhoff manifold is the one of next smallest volume.
The complement in the three-sphere of the figure-eight knot is an arithmetic hyperbolic three-manifold[4] and attains the smallest volume among all cusped hyperbolic three-manifolds.
[5] The Ramanujan conjecture for automorphic forms on
over a number field would imply that for any congruence cover of an arithmetic three-manifold (derived from a quaternion algebra) the spectrum of the Laplace operator is contained in
Many of Thurston's conjectures (for example the virtually Haken conjecture), now all known to be true following the work of Ian Agol,[6] were checked first for arithmetic manifolds by using specific methods.
[7] In some arithmetic cases the Virtual Haken conjecture is known by general means but it is not known if its solution can be arrived at by purely arithmetic means (for instance, by finding a congruence subgroup with positive first Betti number).
Arithmetic manifolds can be used to give examples of manifolds with large injectivity radius whose first Betti number vanishes.
[8][9] A remark by William Thurston is that arithmetic manifolds "...often seem to have special beauty.
"[10] This can be substantiated by results showing that the relation between topology and geometry for these manifolds is much more predictable than in general.