Arithmetic Subgroups and Applications

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.