In Minkowski spacetime, the notion of distance is quite different from the usual Euclidean one. For example, the "length" of a non-zero vector, called its semi-norm, can be positive, zero or negative :
Relativity's requirement that the speed of light be constant, regardless of a free-falling observer's speed, imposes a special structure on spacetime.
So the isometries of Minkowski spacetime, called Lorentz isometries, may seem odd to one accustomed to Euclidean surroundings. And perhaps one should expect groups of Lorentz isometries to display quite different behavior from that of the familiar Euclidean rigid motions.
Yet many were surprised when Margulis unveiled free, non-abelian and discrete groups of Lorentz isometries which act properly discontinuously on three-dimensional spacetime! Contrast this with the Euclidean case, where any cristallographic group must contain a subgroup of translations of finite index.
Schottky groups are free, non-abelian and discrete groups whose elements other than the identity are all hyperbolic. The Margulis examples are affine deformations of Schottky groups, that is, groups of affine isometries whose linear parts are Schottky groups.
The quotient of affine spacetime by such a deformation is called a Margulis spacetime. My thesis is about Margulis spacetimes. I have found new criteria for an affine deformation of a Schottky group to act properly discontinuously on Minkowski spacetime. And the search goes on.
Fundamental polyhedra for Margulis spacetimes may be constructed using crooked planes. These were invented by Todd Drumm because the fundamental domain of a free, non-abelian group must be bounded by objects that are easy to assemble without intersecting each other. The picture above is a truncated view of a crooked plane.