Arithmetic hyperbolic 3 manifolds perfectoid spaces and galois representations i peter scholze
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
Contents
- Arithmetic hyperbolic 3 manifolds perfectoid spaces and galois representations i peter scholze
- Arithmetic hyperbolic 3 manifolds perfectoid spaces and galois representations iii peter scholze
- Quaternion algebras
- Arithmetic Kleinian groups
- Examples
- Trace field of arithmetic manifolds
- Volume formula
- Finiteness results
- Remarkable arithmetic hyperbolic three manifolds
- Spectrum and Ramanujan conjectures
- Arithmetic manifolds in three dimensional topology
- References
Arithmetic hyperbolic 3 manifolds perfectoid spaces and galois representations iii peter scholze
Quaternion algebras
A quaternion algebra over a field
A quaternion algebra is said to be split over
If
Arithmetic Kleinian groups
A subgroup of
The main fact about these groups is that they are discrete subgroups and they have finite covolume for the Haar measure on
An arithmetic Kleinian group is any subgroup of
Examples
Examples are provided by taking
If
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
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:
Volume formula
For the volume an arithmetic three manifold
where
Finiteness results
A consequence of the volume formula in the previous paragraph is that
GivenThis 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
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. Interestingly, 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 (i.e. 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: