Nil Geometry

The Geometry of the Lie Group consisting of Real Matrices of the form

$$\left[{\matrix{1 & x & y\cr 0 & 1 & z\cr 0 & 0 & 1\cr}}\right],$$

i.e., the Heisenberg Group.

See also Heisenberg Group, Lie Group, Thurston's Geometrization Conjecture