Kleinian Groups

Kleinian groups are collections of transformations that act on three-dimensional space or the complex plane, preserving angles and shapes but possibly stretching or twisting them. These groups are used to study complex geometric structures called hyperbolic spaces, which have constant negative curvature. They help mathematicians understand how shapes can be symmetrically repeated or transformed in such spaces, often leading to intricate, fractal-like boundaries. Kleinian groups connect geometry, complex analysis, and topology, offering insights into the fabric of abstract spaces and their symmetries.