Trace field of a representation

In mathematics, the trace field of a linear group is the field generated by the traces of its elements. It is mostly studied for Kleinian and Fuchsian groups, though related objects are used in the theory of lattices in Lie groups, often under the name field of definition.

Fuchsian and Kleinian groups

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Trace field and invariant trace fields for Fuchsian groups

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Fuchsian groups are discrete subgroups of  . The trace of an element in   is well-defined up to sign (by taking the trace of an arbitrary preimage in  ) and the trace field of   is the field generated over   by the traces of all elements of   (see for example in Maclachlan & Reid (2003)).

The invariant trace field is equal to the trace field of the subgroup   generated by all squares of elements of   (a finite-index subgroup of  ).[1]

The invariant trace field of Fuchsian groups is stable under taking commensurable groups. This is not the case for the trace field;[2] in particular the trace field is in general different from the invariant trace field.

Quaternion algebras for Fuchsian groups

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Let   be a Fuchsian group and   its trace field. Let   be the  -subalgebra of the matrix algebra   generated by the preimages of elements of  . The algebra   is then as simple as possible, more precisely:[3]

If   is of the first or second type then   is a quaternion algebra over  .

The algebra   is called the quaternion algebra of  . The quaternion algebra of   is called the invariant quaternion algebra of  , denoted by  . As for trace fields, the former is not the same for all groups in the same commensurability class but the latter is.

If   is an arithmetic Fuchsian group then   and   together are a number field and quaternion algebra from which a group commensurable to   may be derived.[4]

Kleinian groups

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The theory for Kleinian groups (discrete subgroups of  ) is mostly similar as that for Fuchsian groups.[5] One big difference is that the trace field of a group of finite covolume is always a number field.[6]

Trace fields and fields of definition for subgroups of Lie groups

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Definition

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When considering subgroups of general Lie groups (which are not necessarily defined as a matrix groups) one has to use a linear representation of the group to take traces of elements. The most natural one is the adjoint representation. It turns out that for applications it is better, even for groups which have a natural lower-dimensional linear representation (such as the special linear groups  ), to always define the trace field using the adjoint representation. Thus we have the following definition, originally due to Ernest Vinberg,[7] who used the terminology "field of definition".[8]

Let   be a Lie group and   a subgroup. Let   be the adjoint representation of  . The trace field of   is the field:
 

If two Zariski-dense subgroups of   are commensurable then they have the same trace field in this sense.

The trace field for lattices

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Let   be a semisimple Lie group and   a lattice. Suppose further that either   is irreducible and   is not locally isomorphic to  , or that   has no factor locally isomorphic to  . Then local rigidity implies the following result.

The field   is a number field.

Furthermore, there exists an algebraic group   over   such that the group of real points   is isomorphic to   and   is contained in a conjugate of  .[7][9] Thus   is a "field of definition" for   in the sense that it is a field of definition of its Zariski closure in the adjoint representation.

In the case where   is arithmetic then it is commensurable to the arithmetic group defined by  .

For Fuchsian groups the field   defined above is equal to its invariant trace field. For Kleinian groups they are the same if we use the adjoint representation over the complex numbers.[10]

Notes

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  1. ^ Maclachlan & Reid 2003, Chapter 3.3.
  2. ^ Maclachlan & Reid 2003, Example 3.3.1.
  3. ^ Maclachlan & Reid 2003, Theorem 3.2.1.
  4. ^ Maclachlan & Reid 2003, Chapter 8.4.
  5. ^ Maclachlan & Reid 2003, Chapter 3.
  6. ^ Maclachlan & Reid 2003, Theorem 3.1.2.
  7. ^ a b Vinberg 1971.
  8. ^ Margulis 1991, Chapter VIII.
  9. ^ Margulis 1991, Chapter VIII, proposition 3.22.
  10. ^ Maclachlan & Reid 2003, p. 321.

References

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  • Vinberg, Ernest (1971). "Rings of definition of dense subgroups of semisimple linear groups". Izv. Akad. Nauk SSSR Ser. Mat. (in Russian). Vol. 35. pp. 45–55. MR 0279206.
  • Maclachlan, Colin; Reid, Alan (2003). The arithmetic of hyperbolic 3-manifolds. Springer.
  • Margulis, Grigory (1991). Discrete subgroups of semisimple Lie groups. Ergebnisse de Mathematik und ihrer Grenzgebiete. Springer-Verlag. ISBN 3-540-12179-X. MR 1090825.