Arithmetic group

In mathematics, arithmetic groups play an important role in number theory, differential geometry, topology, algebraic geometry and the theory of Lie groups. It is arithmetically defined lattice in Lie groups; classic examples are the modular group and generally the groups for. Arithmetizität is always defined in terms of a surrounding Lie group. By a theorem of Margulis all irreducible lattices are in semi- simple Lie groups of rank without compact factor always arithmetic subgroups.

  • 4.1 gap end Tori
  • 4.2 Q- rank
  • 4.3 Examples
  • 4.4 Geometric Interpretation

Definition

Let be a non-compact semisimple Lie group, a subgroup. is called arithmetic if it

  • Over a defined continuous linear algebraic group and
  • An isomorphism ( for suitable compact normal subgroup )

Are to be so commensurable.

Note: An over -defined linear algebraic group is - by definition - an area defined by polynomials with rational coefficients subgroup. If a is defined linear algebraic group, then a grid is. Consequently, any arithmetic group is a lattice in the connected component of the ambient Lie group.

Examples

  • By definition, it is clear that as well as to commensurate groups are arithmetically.
  • Denote the set of Gaussian integers. is an arithmetic subgroup of, because it is the canonical embedding.
  • Be where the diagonal matrix and is referred. Then, an arithmetic subgroup of, for defined by polynomials with rational coefficients.
  • In the following we want to define a class of less obvious examples apply, namely the Hilbert module groups.

Be

A real quadratic number field - for a square-free integer with - and his whole ring. There are two defined by embeddings and accordingly two embeddings.

We consider the semi- simple Lie group

And want to show that an arithmetic group.

We first consider the algebraic variety

And by

Then is.

We notice that there is a bijective with ( additive and multiplicative ) homomorphism

So for all out there, namely.

Now we consider the linear algebraic group

( Here are 2x2 blocks in a 4x4 matrix. )

We define a group homomorphism

Is actually from after: obviously are the blocks of image matrices in, also is with.

From the bijectivity of follows that also bijective and hence is an isomorphism.

Because demonstrates that the Arithmetizität of.

Arithmetic subgroups of SL (n, R)

All arithmetic subgroups of division algebras means you can construct by means of unitary groups or a combination of these two methods.

Division algebras

Let be a field extension of with and be the totality of the ring. Be with and for the non-trivial element and all.

We consider the division algebra and.

Then an arithmetic subgroup of.

Unitary groups

Be with and is the non-trivial element of the Galois group. Let be a Hermitian matrix.

We consider.

Then an arithmetic subgroup of.

Combination

Be with and is the non-trivial element. Be a division algebra over can be so continued at a Antiautomorphismus of. Let be a Hermitian matrix, that is,.

Then an arithmetic subgroup of.

Q- rank and R- rank

Gap end Tori

Be an algebraic group. A torus is a closed, connected subgroup (above ) is diagonalizable, ie there is a change of basis so that consists of diagonalizable matrices.

The torus - called divisive when you can choose. For example, no - splitting torus is in the group of diagonal matrices ( with determinant 1) yet. The rank of an algebraic group is the maximum dimension of a gap torus. For example, is or.

A torus is called splitting - when it is defined and one can choose.

Q- rank

For an arithmetic group, there is by definition an above defined contiguous linear algebraic group and an isomorphism such that ( modulo compact groups) is isomorphic to the image of. The rank is defined as the dimension of a maximum gap in the torus. ( Note that only depends on, but that various arithmetic subgroups of a Lie group can have different rank, because to be elected algebraic groups differ. )

Examples

It is easily seen that. The arithmetic subgroup has so rank. The rank of the above discussed Hilbert modular group, however, is the rank of the above-constructed group. It can be shown that a maximum cleaving torus is, therefore.

Geometric interpretation

Let be a non-compact semisimple Lie group without compact factor, a maximum compact subgroup and an arithmetic lattice. The Killing form defined on a Riemannian metric, we obtain a symmetric space. The ranking of can be interpreted as the dimension of a maximal flat subspace (ie a simply connected totally geodesic submanifold - with constant sectional curvature ) in.

The quotient is a locally symmetrical space. The ranking of can be interpreted as the maximum dimension of a flat subspace in a finite superposition of, or as the smallest number such that very finite distance of a finite union -dimensional flat subspaces is. In particular, if is compact.

Arithmetizitäts set of Margulis

Theorem: Let a semisimple Lie group with no compact factor. Then each irreducible lattice is arithmetic.

Remarks: A mesh having a discrete subgroup, wherein the volume is calculated with respect to the hair measure. A lattice is called irreducible if there is no decomposition with grids.

Latticework Ring of integers Killing-Form Flat manifold
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