Editing Hyperbolic geometry (section)

==Homogeneous structure==
[[Hyperbolic space]] of dimension ''n'' is a special case of a Riemannian [[symmetric space]] of noncompact type, as it is [[isomorphic]] to the quotient
:: <math>\mathrm{O}(1,n)/(\mathrm{O}(1) \times \mathrm{O}(n)).</math>
The [[orthogonal group]] {{nowrap|O(1, ''n'')}} [[Group action (mathematics)|acts]] by norm-preserving transformations on [[Minkowski space]] '''R'''<sup>1,''n''</sup>, and it acts [[Group action (mathematics)#Types of actions|transitively]] on the two-sheet hyperboloid of norm 1 vectors.  Timelike lines (i.e., those with positive-norm tangents) through the origin pass through antipodal points in the hyperboloid, so the space of such lines yields a model of hyperbolic ''n''-space.  The [[Stabilizer subgroup|stabilizer]] of any particular line is isomorphic to the [[Direct product of groups|product]] of the orthogonal groups O(''n'') and O(1), where O(''n'') acts on the tangent space of a point in the hyperboloid, and O(1) reflects the line through the origin.  Many of the elementary concepts in hyperbolic geometry can be described in [[linear algebra]]ic terms: geodesic paths are described by intersections with planes through the origin, dihedral angles between hyperplanes can be described by inner products of normal vectors, and hyperbolic reflection groups can be given explicit matrix realizations.

In small dimensions, there are exceptional isomorphisms of [[Lie group]]s that yield additional ways to consider symmetries of hyperbolic spaces.  For example, in dimension 2, the isomorphisms {{nowrap|SO<sup>+</sup>(1, 2) ≅ PSL(2, '''R''') ≅ PSU(1, 1)}} allow one to interpret the upper half plane model as the quotient {{nowrap|SL(2, '''R''')/SO(2)}} and the Poincaré disc model as the quotient {{nowrap|SU(1, 1)/U(1)}}.  In both cases, the symmetry groups act by fractional linear transformations, since both groups are the orientation-preserving stabilizers in {{nowrap|PGL(2, '''C''')}} of the respective subspaces of the Riemann sphere.  The Cayley transformation not only takes one model of the hyperbolic plane to the other, but realizes the isomorphism of symmetry groups as conjugation in a larger group.  In dimension 3, the fractional linear action of {{nowrap|PGL(2, '''C''')}} on the Riemann sphere is identified with the action on the conformal boundary of hyperbolic 3-space induced by the isomorphism {{nowrap|O<sup>+</sup>(1, 3) ≅ PGL(2, '''C''')}}. This allows one to study isometries of hyperbolic 3-space by considering spectral properties of representative complex matrices.  For example, parabolic transformations are conjugate to rigid translations in the upper half-space model, and they are exactly those transformations that can be represented by [[unipotent]] [[Triangular matrix|upper triangular]] matrices.