Arithmetic hyperbolic 3-manifold
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. They are particular instances of arithmetic groups. An arithmetic hyperbolic three-manifold is the quotient of hyperbolic space formula_1 by an arithmetic Kleinian group. These manifolds include some particularly beautiful or remarkable examples. Definition and examples. Quaternion algebras. A quaternion algebra over a field formula_2 is a four-dimensional central simple formula_2-algebra. A quaternion algebra has a basis formula_4 where formula_5 and formula_6. A quaternion algebra is said to  be split over formula_2 if it is isomorphic as an formula_2-algebra to the algebra of matrices formula_9; a quaternion algebra over an algebraically closed field is always split. If formula_10 is an embedding of formula_2 into a field formula_12 we shall denote by formula_13 the algebra obtained by extending scalars from formula_2 to formula_12 where we view formula_2 as a subfield of formula_12 via formula_10. Arithmetic Kleinian groups. A subgroup of formula_19 is said to be "derived from a quaternion algebra" if it can be obtained through the following construction. Let formula_2 be a number field which has exactly  two embeddings into formula_21 whose image is not contained in formula_22 (one conjugate to the other). Let formula_23 be a quaternion algebra over formula_2 such that for any embedding formula_25 the algebra formula_26 is isomorphic to the Hamilton quaternions. Next we need an order formula_27 in formula_23. Let formula_29 be the group of elements in formula_27 of reduced norm 1 and let formula_31 be its image in formula_32 via formula_33. We then consider the Kleinian group obtained as the image in formula_19 of formula_35. The main fact about these groups is that they are discrete subgroups and they have finite  covolume for the Haar measure on formula_19. Moreover, the construction above yields a cocompact subgroup if and only if the algebra formula_23 is not split over formula_2. The discreteness is a rather immediate consequence of the fact that formula_23 is only split at its complex embeddings. The finiteness of covolume is harder to prove. An "arithmetic Kleinian group" is any subgroup of formula_19 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 formula_19). Examples. Examples  are provided by taking formula_2 to be an imaginary quadratic field, formula_43 and formula_44 where formula_45 is the ring of integers of formula_2 (for example formula_47 and formula_48). The groups thus obtained are the Bianchi groups. They are not cocompact, and any arithmetic Kleinian group which is not commensurable to a conjugate of a Bianchi group is cocompact. If formula_23 is any quaternion algebra over an imaginary quadratic number field formula_2 which is not isomorphic to a matrix algebra then the unit groups of orders in formula_23 are cocompact. Trace field of arithmetic manifolds. 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 formula_2 the invariant trace field equals formula_2. 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: Geometry and spectrum of arithmetic  hyperbolic three-manifolds. Volume formula. For the volume an arithmetic three manifold formula_59 derived from a maximal order in a quaternion algebra formula_23 over a number field formula_61 we have the expression: formula_62 where formula_63 are the discriminants of formula_64 respectively, formula_65 is the Dedekind zeta function of formula_2 and formula_67. Finiteness results. 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. Remarkable arithmetic hyperbolic three-manifolds. The Weeks manifold is the hyperbolic three-manifold of smallest volume 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 and attains the smallest volume among all cusped hyperbolic three-manifolds. Spectrum and Ramanujan conjectures. The Ramanujan conjecture for automorphic forms on formula_68 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 formula_69. Arithmetic manifolds in three-dimensional topology. Many of Thurston's conjectures (for example the virtually Haken conjecture), now all known to be true following the work of Ian Agol, were checked first for arithmetic manifolds by using specific methods. 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. A remark by William Thurston is that arithmetic manifolds "...often seem to have special beauty." This can be substantiated by results showing that the relation between topology and geometry for these manifolds is much more predictable than in general. For example: