Many of us know that a torus can be constructed by gluing together the opposite ends of a parallelogram. Different parallelograms yield geometrically different surfaces. For surfaces of higher genus, with more holes, the surface can be constructed by gluing together hexagons with six right angles (yes that can happen in hyperbolic geometry). Then all possible surfaces arise from the gluing of some set of hexagons. The hexagons are from a "pairs of pants" decomposition of the surface which is the big idea of this talk. Understanding the possible constructions depends on the following proposed Congruence Law in Hyperbolic Geometry: If two right-angled hexagons have three corresponding sides of equal length then they are congruent.

The talk will explain all concepts from the ground up. The proposed congruence law will be related to the familiar side-side-side and side-angle side congruence theorems from high school geometry.