Hyperbolic groups belong to the obscure area of geometric group theory. Their definition (https://en.wikipedia.org/wiki/Hyperbolic_group) is difficult to work through–thin triangles? What are those?
So why should you care? In increasing order of coolness:
- Group theory: Finite groups have been studied for centuries. The next big class after the class of finite groups is infinite, finitely generated groups. The vast majority of such groups are hyperbolic. The word problem, infamously hard to solve and similar to the halting problem) is solvable in hyperbolic groups.
- Cosmology: It is well known that every surface can be made spherical, Euclidean, or hyperbolic. A little less well known is the Geometrization Theorem, which says there are 8 possible geometries of 3-manifolds, of which the largest group (by far!) is the group of hyperbolic universes; that is, universes whose fundamental group is hyperbolic. A random universe is hyperbolic with a very high probability.
- Fractals: This is part of my own research: Every hyperbolic group has a fractal nature at infinity which can often be drawn explicitly by subdivision rules. These subdivision rules carry all of the information about the hyperbolic group. Some images can be seen at http://www.math.vt.edu/people/floyd/subdivisionrules/gallery/gallery.html. The image at the top of this post is a finite subdivision rule.