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In mathematics, a Cannon–Thurston map is any of a number of continuous group-equivariant maps between the boundaries of two hyperbolic metric spaces extending a discrete isometric actions of the group on those spaces. The notion originated from a seminal 1980s preprint of James Cannon and William Thurston "Group-invariant Peano curves" (eventually published in 2007) about fibered hyperbolic 3-manifolds. Cannon–Thurston maps provide many natural geometric examples of space-filling curves.
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