Arithmetic of hyperbolic manifolds.

*(English)*Zbl 0777.57007
Topology ’90, Contrib. Res. Semester Low Dimensional Topol., Columbus/OH (USA) 1990, Ohio State Univ. Math. Res. Inst. Publ. 1, 273-310 (1992).

[For the entire collection see Zbl 0747.00024.]

By a hyperbolic \(3\)-manifold the authors mean a complete orientable hyperbolic \(3\)-manifold of finite volume. Such a manifold \(M\) is the orbit space of the hyperbolic \(3\)-space with respect to the natural action of a discrete subgroup \(\Gamma\) of the orientable isometry group \(\text{PSL}_ 2({\mathbf C})\), \(\Gamma\) is allowed to have torsion, in which case \(M\) is actually a hyperbolic orbifold. A suitable equivalence relation among such \(M\) is commensurability: two \(3\)-manifolds are commensurable if each has a finite covering such that the two total spaces are homeomorphic.

Certain hyperbolic \(3\)-manifolds can be derived, in a certain way, from pairs \((k,A)\), where \(k\) is a number field with exactly one complex place and \(A\) is a quaternion algebra over \(k\) which is ramified at all real Archimedean places of \(k\) [see M.-F. Vignéras, Arithmétique des algèbres de quaternions, Lecture Notes in Mathematics, 800 (1980; Zbl 0422.12008)]. A hyperbolic \(3\)-manifold \(M\) of this sort is said to be arithmetic. There exists a bijective correspondence between pairs \((k,A)\) as above and commensurability classes of arithmetic (hyperbolic \(3\)- dimensional) manifolds. This offers the possibility of using arithmetic tools at investigating arithmetic manifolds and makes arithmetic manifolds easier to handle.

One of the objects of the present paper is to extend the use of arithmetic tools to more general hyperbolic \(3\)-manifolds. In sections 2- 3 the authors assign to each hyperbolic \(3\)-manifold \(M\) and the associated group \(\Gamma\leq \text{PSL}_ 2({\mathbf C})\) a number field \(k(M)=k(\Gamma)\) (viz. the field obtained by adjoining to \({\mathbf Q}\) the traces of all elements of \(\Gamma^{(2)}\), the subgroup of \(\Gamma\) generated by the squares of all elements) and a quaternion algebra \(A(M)=A(\Gamma)\) over it (viz. the rational vector subspace of the matrix algebra \(M_ 2({\mathbf C})\) spanned by the preimage of \(\Gamma^{(2)}\) in \(\text{SL}_ 2({\mathbf C})\)). The pair \((k(M),A(M))\) is a commensurability invariant of \(M\) (in the arithmetic case identical with the pair \((k,A)\) mentioned in the preceding paragraph), but noncommensurable (nonarithmetic) manifolds can have the same such pair. The authors show that for a noncompact \(M\), \(k(M)\) can be derived from any triangulation of \(M\) by ideal tetrahedra and \(A(M)\) is the full \(2\times 2\) matrix algebra over \(k(M)\). If \(M\) admits an orientation reversing symmetry, at least “up to commensurability”, then \(k(M)\) and \(A(M)\) are invariant under complex conjugation. There are more results in sections 2-3, but stating them would require additional definitions.

In section 4 the authors obtain a lower bound of lengths of geodesics in noncompact arithmetic orbifolds. Half of their main theorem of this section states that if a cusped arithmetic orbifold \(M\) contains a geodesic of length less than \(\text{Arcosh}(3/2)\), then \(M\) is commensurable with one of seven explicitly given orbifolds and the length of the geodesic in question belongs to a given list of 18 numbers. There is also a universal lower bound of lengths of geodesics in closed arithmetic orbifolds provided the classical Lehmer conjecture of number theory is correct.

In most of the rest of the paper (sections 5-8) the authors study links in \(S^ 3\) which have the form of a circular chain of consecutively and simply linked unknotted circles and the chain is allowed to have any number of half-twists. The authors find out precisely when the complements of two such links are commensurable, when they are hyperbolic, and when they are arithmetic (and they also decide some other questions). As an essential tool, they investigate generalized hyperbolic Dehn surgery on one cusp of the complement of the Whitehead link.

In the short concluding sections 9 and 10 the authors pose some arithmetic questions about hyperbolic knot complements.

By a hyperbolic \(3\)-manifold the authors mean a complete orientable hyperbolic \(3\)-manifold of finite volume. Such a manifold \(M\) is the orbit space of the hyperbolic \(3\)-space with respect to the natural action of a discrete subgroup \(\Gamma\) of the orientable isometry group \(\text{PSL}_ 2({\mathbf C})\), \(\Gamma\) is allowed to have torsion, in which case \(M\) is actually a hyperbolic orbifold. A suitable equivalence relation among such \(M\) is commensurability: two \(3\)-manifolds are commensurable if each has a finite covering such that the two total spaces are homeomorphic.

Certain hyperbolic \(3\)-manifolds can be derived, in a certain way, from pairs \((k,A)\), where \(k\) is a number field with exactly one complex place and \(A\) is a quaternion algebra over \(k\) which is ramified at all real Archimedean places of \(k\) [see M.-F. Vignéras, Arithmétique des algèbres de quaternions, Lecture Notes in Mathematics, 800 (1980; Zbl 0422.12008)]. A hyperbolic \(3\)-manifold \(M\) of this sort is said to be arithmetic. There exists a bijective correspondence between pairs \((k,A)\) as above and commensurability classes of arithmetic (hyperbolic \(3\)- dimensional) manifolds. This offers the possibility of using arithmetic tools at investigating arithmetic manifolds and makes arithmetic manifolds easier to handle.

One of the objects of the present paper is to extend the use of arithmetic tools to more general hyperbolic \(3\)-manifolds. In sections 2- 3 the authors assign to each hyperbolic \(3\)-manifold \(M\) and the associated group \(\Gamma\leq \text{PSL}_ 2({\mathbf C})\) a number field \(k(M)=k(\Gamma)\) (viz. the field obtained by adjoining to \({\mathbf Q}\) the traces of all elements of \(\Gamma^{(2)}\), the subgroup of \(\Gamma\) generated by the squares of all elements) and a quaternion algebra \(A(M)=A(\Gamma)\) over it (viz. the rational vector subspace of the matrix algebra \(M_ 2({\mathbf C})\) spanned by the preimage of \(\Gamma^{(2)}\) in \(\text{SL}_ 2({\mathbf C})\)). The pair \((k(M),A(M))\) is a commensurability invariant of \(M\) (in the arithmetic case identical with the pair \((k,A)\) mentioned in the preceding paragraph), but noncommensurable (nonarithmetic) manifolds can have the same such pair. The authors show that for a noncompact \(M\), \(k(M)\) can be derived from any triangulation of \(M\) by ideal tetrahedra and \(A(M)\) is the full \(2\times 2\) matrix algebra over \(k(M)\). If \(M\) admits an orientation reversing symmetry, at least “up to commensurability”, then \(k(M)\) and \(A(M)\) are invariant under complex conjugation. There are more results in sections 2-3, but stating them would require additional definitions.

In section 4 the authors obtain a lower bound of lengths of geodesics in noncompact arithmetic orbifolds. Half of their main theorem of this section states that if a cusped arithmetic orbifold \(M\) contains a geodesic of length less than \(\text{Arcosh}(3/2)\), then \(M\) is commensurable with one of seven explicitly given orbifolds and the length of the geodesic in question belongs to a given list of 18 numbers. There is also a universal lower bound of lengths of geodesics in closed arithmetic orbifolds provided the classical Lehmer conjecture of number theory is correct.

In most of the rest of the paper (sections 5-8) the authors study links in \(S^ 3\) which have the form of a circular chain of consecutively and simply linked unknotted circles and the chain is allowed to have any number of half-twists. The authors find out precisely when the complements of two such links are commensurable, when they are hyperbolic, and when they are arithmetic (and they also decide some other questions). As an essential tool, they investigate generalized hyperbolic Dehn surgery on one cusp of the complement of the Whitehead link.

In the short concluding sections 9 and 10 the authors pose some arithmetic questions about hyperbolic knot complements.

Reviewer: J.Vrabec (Ljubljana)

##### MSC:

57N10 | Topology of general \(3\)-manifolds (MSC2010) |

30F40 | Kleinian groups (aspects of compact Riemann surfaces and uniformization) |

20H10 | Fuchsian groups and their generalizations (group-theoretic aspects) |

57M25 | Knots and links in the \(3\)-sphere (MSC2010) |

11F06 | Structure of modular groups and generalizations; arithmetic groups |

57S30 | Discontinuous groups of transformations |