Arithmetic group

In mathematics , arithmetic groups play an important role in number theory , differential geometry , topology , algebraic geometry and in the theory of Lie groups . These are arithmetically defined grids in Lie groups ; classic examples are the module group and generally the groups for . Arithmeticity is always defined in terms of a surrounding Lie group. According to one of Margulis' theorem , all irreducible lattices in semi-simple Lie groups of rank without a compact factor are always arithmetic subgroups. ${\displaystyle SL(2,\mathbb {Z} )}$ ${\displaystyle SL(n,\mathbb {Z} )}$ ${\displaystyle n\geq 2}$ ${\displaystyle \geq 2}$

definition

Let be a non-compact semi - simple Lie group , a subgroup . is called arithmetic if it is ${\displaystyle G}$ ${\displaystyle \Gamma \subset G}$ ${\displaystyle \Gamma }$

• an over defined connected linear algebraic group and ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathrm {G} \subset GL(n,\mathbb {C} )}$
• an isomorphism (for suitable compact normal divisors ) ${\displaystyle \phi :G/K\rightarrow \mathrm {G} (\mathbb {R} )_{0}/K^{\prime }}$ ${\displaystyle K,K^{\prime }}$

are so commensurable to be. ${\displaystyle \phi (\Gamma K)}$ ${\displaystyle (\mathrm {G} (\mathbb {Z} )\cap \mathrm {G} (\mathbb {R} )_{0})K^{\prime }}$

Note: An over defined linear algebraic group is - by definition - a subgroup defined by polynomials with rational coefficients . If is an over defined linear algebraic group, then by the theorem of Borel and Harish-Chandra a lattice is in . Hence, each arithmetic group is a lattice in the connected component of the surrounding Lie group . ${\displaystyle \mathbb {Q} }$ ${\displaystyle G\subset GL(n,\mathbb {R} )}$ ${\displaystyle \mathrm {G} }$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathrm {G} (\mathbb {Z} )}$ ${\displaystyle \mathrm {G} (\mathbb {R} )}$

Examples

• By definition, it is clear that and groups that are too commensurate are arithmetic. ${\displaystyle SL(n,\mathbb {Z} )\subset SL(n,\mathbb {R} )}$ ${\displaystyle SL(n,\mathbb {Z} )}$

• Denote the group of whole Gaussian numbers . is an arithmetic subgroup of , because it is for canonical embedding . ${\displaystyle \mathbb {Z} \left[i\right]\subset \mathbb {C} }$ ${\displaystyle GL(n,\mathbb {Z} \left[i\right])}$ ${\displaystyle GL(n,\mathbb {C} )}$ ${\displaystyle \phi (GL(n,\mathbb {Z} \left[i\right])=\phi (GL(n,\mathbb {C} ))\cap GL(2n,\mathbb {Z} )}$ ${\displaystyle \phi :GL(n,\mathbb {C} )\rightarrow GL(2n,\mathbb {R} )}$

• Let , where denotes the diagonal matrix and let . Then is an arithmetic subgroup of , because is defined by polynomials with rational coefficients. ${\displaystyle SO(1,n)=\left\{g\in SL(n+1,\mathbb {R} ):gI_{1,n}g^{T}=I_{1,n}\right\}}$ ${\displaystyle I_{1,n}}$ ${\displaystyle I_{1,n}=diag(1,-1,-1,\ldots ,-1)}$ ${\displaystyle SO(1,n,\mathbb {Z} )=SO(1,n)\cap SL(n+1,\mathbb {Z} )}$ ${\displaystyle SO(1,n,\mathbb {Z} )}$ ${\displaystyle SO(1,n)}$ ${\displaystyle SO(1,n)}$

• In the following we want to apply the definition to a class of less obvious examples, namely to Hilbert's module groups .

Be

${\displaystyle k=Q\left[{\sqrt {D}}\right]=\left\{a+b{\sqrt {D}}:a,b\in \mathbb {Q} \right\}}$

a real square number field - for a square-free integer with - and its integral ring . There are two defined embeddings and, accordingly, two embeddings . ${\displaystyle D>0}$ ${\displaystyle D\cong 3\ mod\ 4}$ ${\displaystyle O_{k}\subset k}$ ${\displaystyle \sigma _{\pm }(a+b{\sqrt {D}})=a\pm b{\sqrt {D}}}$ ${\displaystyle \sigma _{\pm }:k\rightarrow \mathbb {R} }$ ${\displaystyle \sigma _{\pm }:SL(2,k)\rightarrow SL(2,\mathbb {R} )}$

We consider the semi-simple Lie group

${\displaystyle G=SL(2,\mathbb {R} )\times SL(2,\mathbb {R} )}$

and the subgroup

${\displaystyle \Gamma =\left\{(\sigma _{+}(A),\sigma _{-}(A)):A\in SL(2,O_{k})\right\}\subset G}$

and want to show that is an arithmetic group. ${\displaystyle \Gamma }$

We first consider the algebraic variety

${\displaystyle \mathrm {H} :=\left\{{\begin{pmatrix}a&b\\c&d\end{pmatrix}}:d=a,c=bD\right\}\subset Mat(2,\mathbb {C} )}$

and through

${\displaystyle \psi (a+b{\sqrt {D}})={\begin{pmatrix}a&b\\bD&a\end{pmatrix}}}$defined homomorphism  : . ${\displaystyle \psi :k\rightarrow \mathrm {H} (\mathbb {Q} )\subset Mat(2,\mathbb {Q} )}$

Then is . ${\displaystyle \psi (O_{k})=\mathrm {H} (\mathbb {Z} )}$

We notice that there is a bijective (additive and multiplicative) homomorphism with ${\displaystyle \Psi :\mathbb {R} \times \mathbb {R} \rightarrow \mathrm {H} (\mathbb {R} )}$

${\displaystyle \Psi \circ (\sigma _{+},\sigma _{-})=\psi }$,

so there for all , namely . ${\displaystyle \Psi (a+b{\sqrt {D}},a-b{\sqrt {D}})=\psi (a+b{\sqrt {D}})}$ ${\displaystyle a+b{\sqrt {D}}\in k}$ ${\displaystyle \Psi (x,y)={\begin{pmatrix}{\frac {x+y}{2}}&{\frac {x-y}{2{\sqrt {D}}}}\\{\frac {x-y}{2}}{\sqrt {D}}&{\frac {x+y}{2}}\end{pmatrix}}}$

Now let's consider the linear algebraic group

${\displaystyle \mathrm {G} :=\left\{X={\begin{pmatrix}A&B\\C&D\end{pmatrix}}:A,B,C,D\in \mathrm {H} ,AD-BC={\begin{pmatrix}1&0\\0&1\end{pmatrix}}\right\}\subset GL(4,\mathbb {C} )}$.

(Here are 2x2 blocks in a 4x4 matrix.) ${\displaystyle A,B,C,D}$

We define a group homomorphism

${\displaystyle \phi :SL(2,\mathbb {R} )\times SL(2,\mathbb {R} )\rightarrow \mathrm {G} (\mathbb {R} )\subset GL(4,\mathbb {R} )}$through . ${\displaystyle \phi \left({\begin{pmatrix}a&b\\c&d\end{pmatrix}},{\begin{pmatrix}a^{\prime }&b^{\prime }\\c^{\prime }&d^{\prime }\end{pmatrix}}\right)={\begin{pmatrix}\Psi (a,a^{\prime })&\Psi (b,b^{\prime })\\\Psi (c,c^{\prime })&\Psi (d,d^{\prime })\end{pmatrix}}}$

${\displaystyle \phi }$actually reproduces: obviously the blocks of the image matrices are in , and is with . ${\displaystyle \mathrm {G} (\mathbb {R} )}$ ${\displaystyle \mathrm {H} }$ ${\displaystyle \Psi (a,a^{\prime })\Psi (d,d^{\prime })-\Psi (b,b^{\prime })\Psi (c,c^{\prime })={\begin{pmatrix}{\frac {\det(X)+\det(Y)}{2}}&{\frac {\det(X)-\det(Y)}{2{\sqrt {D}}}}\\{\frac {\det(X)-\det(Y)}{2}}{\sqrt {D}}&{\frac {\det(X)+\det(Y)}{2}}\end{pmatrix}}={\begin{pmatrix}1&0\\0&1\end{pmatrix}}}$ ${\displaystyle X={\begin{pmatrix}a&b\\c&d\end{pmatrix}},Y={\begin{pmatrix}a^{\prime }&b^{\prime }\\c^{\prime }&d^{\prime }\end{pmatrix}}}$

From the bijectivity of it follows that it is also bijective and therefore an isomorphism. ${\displaystyle \Psi }$ ${\displaystyle \phi }$

Because of this proves the arithmeticity of . ${\displaystyle \phi (\Gamma )=\mathrm {G} (\mathbb {Z} )}$ ${\displaystyle \Gamma }$

Arithmetic subgroups of SL (n, R)

All arithmetic sub-groups of one can by means of division algebras , by means of unitary groups construct or by a combination of these two methods. ${\displaystyle SL(n,\mathbb {R} )}$

Division algebras

Be a body extension of with and be the wholeness ring of . Be with and for the nontrivial element and all . ${\displaystyle \mathbb {Q} }$ ${\displaystyle \left[F:\mathbb {Q} \right]=d}$ ${\displaystyle {\mathcal {O}}}$ ${\displaystyle F}$ ${\displaystyle j\in \mathbb {C} }$ ${\displaystyle j^{d}\in \mathbb {Z} }$ ${\displaystyle jx=\eta (x)j}$ ${\displaystyle \eta \in Gal(F/\mathbb {Q} )}$ ${\displaystyle x\in F}$

We consider the division algebra and . ${\displaystyle D=F+Fj+Fj^{2}+\ldots +Fj^{d-1}}$ ${\displaystyle D_{\mathbb {Z} }={\mathcal {O}}+{\mathcal {O}}j+{\mathcal {O}}j^{2}+\ldots +{\mathcal {O}}j^{d-1}}$

Then is an arithmetic subgroup of . ${\displaystyle SL(n,D_{\mathbb {Z} })}$ ${\displaystyle SL(dn,\mathbb {R} )}$

Unitary groups

Be with and be the nontrivial element of the Galois group . Be a Hermitian matrix . ${\displaystyle F=\mathbb {Q} \left[{\sqrt {r}}\right]}$ ${\displaystyle r\in \mathbb {Z} }$ ${\displaystyle \eta \in Gal(F/\mathbb {Q} )}$ ${\displaystyle A\in GL(n,F)}$

We look at . ${\displaystyle SU(A,\eta ;F)=\left\{g\in SL(n,F):gA(\eta (g))^{T}=A\right\}}$

Then is an arithmetic subgroup of . ${\displaystyle SU(A,\eta ;\mathbb {Z} \left[{\sqrt {r}}\right])=SU(A,\eta ;F)\cap SL(n,\mathbb {Z} \left[{\sqrt {r}}\right])}$ ${\displaystyle SL(n,\mathbb {C} )}$

combination

Be with and be the nontrivial element. Let be a division algebra over such that an antiautomorphism of can be continued. Let be a Hermitian matrix, i.e. H. . ${\displaystyle F=\mathbb {Q} \left[{\sqrt {r}}\right]}$ ${\displaystyle r\in \mathbb {Z} }$ ${\displaystyle \eta \in Gal(F/\mathbb {Q} )}$ ${\displaystyle D}$ ${\displaystyle F}$ ${\displaystyle \eta }$ ${\displaystyle D}$ ${\displaystyle A\in GL(n,D)}$ ${\displaystyle (\eta (A))^{T}=A}$

Then is an arithmetic subgroup of . ${\displaystyle SU(A,\eta ;D_{\mathbb {Z} })}$ ${\displaystyle SU(A,\eta ;D\otimes _{\mathbb {Q} }\mathbb {R} )}$

Q rank and R rank

Divisive tori

Be an algebraic group. A torus is a closed, coherent subgroup , which is (over ) diagonalizable, that is, there is a base change , so that it consists of diagonalizable matrices. ${\displaystyle G\subset GL(n,\mathbb {C} )}$ ${\displaystyle T\subset G}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle B\in GL(n,\mathbb {C} )}$ ${\displaystyle BTB^{-1}}$

The torus is called splitting, if one can choose. For example there is no -splitting torus in , but the group of diagonal matrices (with determinant 1) is. The rank of an algebraic group is the maximum dimension of a -splitting torus. For example is or . ${\displaystyle \mathbb {R} }$ ${\displaystyle B\in GL(n,\mathbb {R} )}$ ${\displaystyle SO(2)}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle SL(2,\mathbb {R} )}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {R} -rk(SL(n,\mathbb {R} ))=n-1}$ ${\displaystyle \mathbb {R} -rk(SO(n))=0}$

A torus is called splitting if it is defined by and one can choose. ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle B\in GL(n,\mathbb {Q} )}$

Q rank

For an arithmetic group there is by definition an over defined, connected linear algebraic group and an isomorphism , so that (modulo compact groups) the image is too isomorphic. The rank of is defined as the dimension of a maximum -cleaving torus in . (Note that only depends on, but that different arithmetic subgroups of a Lie group can have different ranks because the algebraic groups to be selected are different.) ${\displaystyle \Gamma \subset G}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathrm {G} }$ ${\displaystyle \phi }$ ${\displaystyle \Gamma }$ ${\displaystyle \mathrm {G} (\mathbb {Z} )\cap \mathrm {G} (\mathbb {R} )_{0}}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle \Gamma }$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathrm {G} }$ ${\displaystyle \mathbb {Q} -rk(\Gamma )}$ ${\displaystyle \mathrm {G} }$ ${\displaystyle \Gamma _{1},\Gamma _{2}\subset G}$ ${\displaystyle G}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathrm {G} _{1},\mathrm {G} _{2}}$

Examples

It's easy to see that . So the arithmetic subgroup has -range . On the other hand, the rank of the Hilbert module group discussed above is the rank of the group constructed above . One can show that a maximal -cleaving torus is in, hence . ${\displaystyle \mathbb {R} -rk(SL(2,\mathbb {R} )\times SL(2,\mathbb {R} ))=\mathbb {Q} -rk(SL(2,\mathbb {R} )\times SL(2,\mathbb {R} ))=2}$ ${\displaystyle SL(2,\mathbb {Z} )\times SL(2,\mathbb {Z} )\subset SL(2,R)\times SL(2,\mathbb {R} )}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle 2}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathrm {G} :=\left\{X={\begin{pmatrix}A&B\\C&D\end{pmatrix}}:A,B,C,D\in \mathrm {H} ,\det(X)=1\right\}\subset GL(4,\mathbb {C} )}$ ${\displaystyle \left\{diag(\lambda ,\lambda ,{\frac {1}{\lambda }},{\frac {1}{\lambda }}):\lambda \in \mathbb {R} ^{\times }\right\}}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathrm {G} (\mathbb {R} )}$ ${\displaystyle \mathbb {Q} -rk(\Gamma )=\mathbb {Q} -rk(\mathrm {G} (\mathbb {R} ))=1}$

Geometric interpretation

Let be a non-compact semi-simple Lie group with no compact factor, a maximally compact subgroup, and an arithmetic lattice. The Killing form defines a Riemannian metric on , one obtains a symmetrical space . The rank of can be interpreted as the dimension of a maximum flat subspace (i.e. a simply connected total geodetic submanifold with constant intersection curvature ) in . ${\displaystyle G}$ ${\displaystyle K}$ ${\displaystyle \Gamma \subset G}$ ${\displaystyle G/K}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle G}$ ${\displaystyle 0}$ ${\displaystyle G/K}$

The quotient is a locally symmetrical space . The rank of can be interpreted as the maximum dimension of a flat subspace in a finite superposition of or as the smallest number so that it is entirely at a finite distance from a finite union of -dimensional flat subspaces. In particular, if is compact. ${\displaystyle X=\Gamma \backslash G/K}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle \Gamma }$ ${\displaystyle X}$ ${\displaystyle r}$ ${\displaystyle X}$ ${\displaystyle r}$ ${\displaystyle \mathbb {Q} -rk(\Gamma )=0}$ ${\displaystyle X}$

Characterization of arithmetic lattices

Theorem ( Margulis ) : An irreducible lattice in a semi-simple Lie group is arithmetic if and only if has an infinite index in its commensurator , i.e. if . ${\displaystyle \Gamma \subset G}$ ${\displaystyle G}$ ${\displaystyle \Gamma }$ ${\displaystyle [\operatorname {comm} _{G}(\Gamma ):\Gamma ]=\infty }$

Margulis' theorem of arithmetic

Theorem : Let be a semi-simple Lie group without a compact factor with . Then every irreducible lattice is arithmetic. ${\displaystyle G}$ ${\displaystyle \mathbb {R} -rk(G)\geq 2}$ ${\displaystyle \Gamma \subset G}$

Explanations: A grid is a discrete subgroup with , where the volume is calculated in relation to the hair measurement . A lattice is called irreducible if there is no decomposition with lattices . ${\displaystyle \Gamma \subset G}$ ${\displaystyle vol(\Gamma \backslash G)<\infty }$ ${\displaystyle G=G_{1}\times G_{2},\Gamma =\Gamma _{1}\times \Gamma _{2}}$ ${\displaystyle \Gamma _{1}\subset G_{1},\Gamma _{2}\subset G_{2}}$

Margulis proved this theorem as a consequence of the superstariness theorem which he proved .

literature

• Lizhen Ji: Arithmetic groups and their generalizations. What, why, and how (= Studies in Advanced Mathematics. Vol. 43). American Mathematical Society, Providence RI 2008, ISBN 978-0-8218-4675-9 .
• Vladimir Platonov , Andrei Rapinchuk: Algebraic Groups and Number Theory (= Pure and Applied Mathematics. Vol. 139). Academic Press, Boston MA et al.1994 , ISBN 0-12-558180-7 , digitized version (PDF; 22.46 MB) .

Web links

• Witte Morris, Dave: Introduction to Arithmetic Groups
• Witte Morris, Dave: Introduction to Arithmetic Groups (slides from a lecture series; PDF; 537 kB)

Individual evidence

1. ^ Margulis, GA: Arithmeticity of the irreducible lattices in the semisimple groups of rank greater than 1. Invent. Math (1984) 76-93. Doi: 10.1007 / BF01388494