Kleinian Groups
There are many natural examples of space-filling, or rather sphere-filling, curves in the theory of doubly degenerate Kleinian groups. For example, Cannon & Thurston (2007) showed that the circle at infinity of the universal cover of a fiber of a mapping torus of a pseudo-Anosov map is a sphere-filling curve. (Here the sphere is the sphere at infinity of hyperbolic 3-space.)
Read more about this topic: Space-filling Curve
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