Arithmetic hyperbolic 3-manifold explained
by an arithmetic Kleinian group.
Definition and examples
Quaternion algebras
See main article: Quaternion algebra.
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 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,
and
where
is the
ring of integers of
(for example
and
). 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
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
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
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:
- Its invariant trace field is a number field with exactly one complex place;
- The traces of its elements are algebraic integers;
- For any
in the group,
and any embedding
we have
.
Geometry and spectrum of arithmetic hyperbolic three-manifolds
Volume formula
For the volume an arithmetic three manifold
derived from a maximal order in a quaternion algebra
over a number field
we have the expression:
where
are the
discriminants of
respectively,
is the
Dedekind zeta function of
and
.
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[1] 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[2] and attains the smallest volume among all cusped hyperbolic three-manifolds.[3]
Spectrum and Ramanujan conjectures
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
.
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,[4] were checked first for arithmetic manifolds by using specific methods.[5] 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.[6] [7]
A remark by William Thurston is that arithmetic manifolds "...often seem to have special beauty."[8] 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:
- For a given genus g there are at most finitely many arithmetic (congruence) hyperbolic 3-manifolds which fiber over the circle with a fiber of genus g.[9]
- There are at most finitely many arithmetic (congruence) hyperbolic 3-manifolds with a given Heegaard genus.[10]
Notes and References
- Milley . Peter . Minimum volume hyperbolic 3-manifolds. . Journal of Topology . 2 . 2009 . 181–192 . 2499442 . 10.1112/jtopol/jtp006. 0809.0346 . 3095292 .
- Riley . Robert . A quadratic parabolic group . Math. Proc. Cambridge Philos. Soc. . 77 . 2 . 1975 . 281–288 . 0412416 . 10.1017/s0305004100051094. 1975MPCPS..77..281R .
- Cao . Chun . Meyerhoff . G. Robert . The orientable cusped hyperbolic 3-manifolds of minimum volume . Invent. Math. . 146 . 3 . 2001 . 451–478 . 1869847 . 10.1007/s002220100167. 2001InMat.146..451C . 123298695 .
- Ian . Agol . Ian Agol. The virtual Haken conjecture . With an appendix by Ian Agol, Daniel Groves, and Jason Manning . Documenta Mathematica . 2013 . 1045–1087 . 18 . 3104553 .
- Lackenby . Marc . Long . Darren D. . Reid . Alan W. . Covering spaces of arithmetic 3-orbifolds . . 2008 . 2008. 2426753. 10.1093/imrn/rnn036. math/0601677 .
- Calegari. Frank . Dunfield . Nathan . Automorphic forms and rational homology 3-spheres . . 10 . 2006 . 295–329 . 10.2140/gt.2006.10.295. math/0508271. 2224458 . 5506430 .
- Boston . Nigel . Ellenberg . Jordan . Jordan Ellenberg. Pro-p groups and towers of rational homology spheres . . 10 . 2006 . 331–334 . 10.2140/gt.2006.10.331. 0902.4567. 2224459 . 14889934 .
- Thurston . William . William Thurston. 1982 . Three-dimensional manifolds, Kleinian groups and hyperbolic geometry . . 6 . 3 . 357–381 . 10.1090/s0273-0979-1982-15003-0. free .
- Biringer . Ian . Souto . Juan . A finiteness theorem for hyperbolic 3-manifolds . J. London Math. Soc. . Second Series . 84 . 227–242 . 2011 . 10.1112/jlms/jdq106. 0901.0300 . 11488751 .
- Gromov . Misha . Mikhail Leonidovich Gromov . Guth . Larry . Larry Guth . Generalizations of the Kolmogorov-Barzdin embedding estimates . Duke Math. J. . 161 . 2549–2603 . 2012 . 13 . 10.1215/00127094-1812840. 1103.3423 . 7295856 .