Bianchi groups: new knowl collection

Core idea

This index links all 39 new concept and theorem knowls in the Bianchi-group collection. Start with Bianchi group, then open its prerequisites as needed. The list below groups the new material from number theory through hyperbolic geometry to arithmetic classification.

Suggested reading path

1. Imaginary quadratic fields → rings of integers → Bianchi groups.
2. Kleinian groups → hyperbolic orbifolds → Bianchi orbifolds.
3. Cusps and ideal classes, then the volume formula.
4. Quaternion algebras → arithmetic Kleinian groups → noncocompact classification.

Number-theory foundations

• Square-free integer
• Imaginary quadratic field
• Algebraic integer
• Ring of integers of a number field
• Ring of integers of an imaginary quadratic field
• Integral basis of a number field
• Discriminant of a number field
• Fractional ideal
• Ideal class group of a number field
• Class number of a number field
• Absolute norm of an ideal
• Dedekind zeta function

Matrices and the Bianchi group

• Special linear group over a ring
• Projective special linear group over a ring
• Bianchi group

Hyperbolic geometry, cusps, and volume

• Kleinian group
• Smooth effective orbifold
• Hyperbolic three-orbifold
• Bianchi orbifold
• Lattice in a Lie group
• Uniform lattice
• Bianchi groups are nonuniform lattices
• Horosphere in hyperbolic three-space
• Horoball in hyperbolic three-space
• Parabolic isometry of hyperbolic three-space
• Cusp of a finite-volume hyperbolic three-orbifold
• Bianchi cusps and ideal classes
• Volume formula for a Bianchi orbifold

Quaternion algebras and arithmetic classification

• Commensurable subgroups
• Quaternion algebra over a field
• Standard conjugation of a quaternion algebra
• Reduced norm in a quaternion algebra
• Split quaternion algebra
• Order in a quaternion algebra
• Norm-one group of a quaternion order
• Ramification of a quaternion algebra at a real place
• Arithmetic Kleinian group
• Arithmeticity of Bianchi groups
• Noncocompact arithmetic Kleinian groups

Existing foundations to reuse

• Number field, integral element, and Dedekind domain.
• Discriminant of a field basis and field norm.
• Discrete subgroup, hyperbolic three-space, and its PSL₂(ℂ) action.
• Group action, orbit-space topology, and subgroup index.

Conventions to watch

This collection uses PSL, the full ring of integers, and hyperbolic curvature −1. The ring-valued SL/PSL companions extend the available vocabulary without changing the existing Lie-group definitions. Bianchi identity and Bianchi classification concern different subjects.

References

1. Bianchi group on Wikipedia, the starting topic inventory. Individual knowls give mathematical references.