Abstract
Arithmetic subgroups are an important source of discrete groups acting freely on manifolds. We need to know that there exist many torsion-free 푺푺L(ퟐퟐ,ℝ) is an “arithmetic” subgroup of 푺푺L(ퟐퟐ,ℝ). The other arithmetic subgroups are not as obvious, but they can be constructed by using quaternion algebras. Replacing the quaternion algebras with larger division algebras yields many arithmetic subgroups of 푺푺L(풏풏,ℝ), with 풏풏>2. In fact, a calculation of group cohomology shows that the only other way to construct arithmetic subgroups of 푺푺L(풏풏, ℝ) is by using arithmetic groups. In this paper justifies Commensurable groups, and some definitions and examples,ℝ-forms of classical simple groups over ℂ, calculating the complexification of each classical group, Applications to manifolds. Let us start with 푺푺푺푺(푛푛,ℂ). This is already a complex Lie group, but we can think of it as a real Lie group of twice the dimension. As such, it has a complexification.
Highlights
In This paper we will give a quite explicit description of the arithmetic subgroups of almost every classical Lie group GG
We wish to find all the possibilities for the group ρρ(GG C)R = ρρ(GG C) ∩ SSSS(NN, R) that can be obtained by considering all the possible choices ofρρ.Let σσdenote complex conjugation, the nontrivial Galois automorphism of C over R
Znn22 is a discrete subset of Rnn22, i.e., every point of Znn22 has an open neighbourhood(for the real topology) containing no other point of Znn22, GGLLnn (Z) is discrete in GGLLnn (R) and it follows that every arithmetic subgroup Γ of a group GGis discrete in GG(R)Let GGbe an algebraic group over Q
Summary
In This paper we will give a quite explicit description of the arithmetic subgroups of almost every classical Lie group GG. There is an exception to this rule, some 8-dimensional orthogonal groups have Q-forms of so-called triality type, that are not classical and will not bediscussed in any detail here given GG, which is a Lie group over R, we would like to know all of itsQ-forms (because, by definition, arithmetic groups are made from Q-forms) [1,2,3]. There are uncountably many lattices in SSSS22(R) (with theassociated locally symmetric spaces being nothing other than Riemann surfaces), but only countably many of them are arithmetic. In higher rank Lie groups, there is the following truly
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