Research
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 Γ°ΒβΒΊΓ°ΒβΒΊ(Γ°ΒΕΈΒ,Γ’βΒ€) is an Γ’β¬ΕarithmeticΓ’β¬Β subgroup of Γ°ΒβΒΊΓ°ΒβΒΊ(Γ°ΒΕΈΒ,Γ’βΒ). 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 Γ°ΒβΒΊΓ°ΒβΒΊ(Γ°ΒβΒ,Γ’βΒ), with Γ°ΒβΒΓ°ΒβΒ>2. In fact, a calculation of group cohomology shows that the only other way to construct arithmetic subgroups of Γ°ΒβΒΊΓ°ΒβΒΊ(Γ°ΒβΒ,Γ’βΒ) is by using arithmetic groups. In this paper justifies Commensurable groups, and some definitions and examples,Γ’βΒ-forms of classical simple groups over Γ’ββ, calculating the complexification of each classical group, Applications to manifolds. Let us start with Γ°ΒβΒΊΓ°ΒβΒΊ(Γ°ΒββΊΓ°ΒββΊ,Γ’ββ). 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.
