The complement of the figure-eight knot geometrically bounds

Abstract

We show that some hyperbolic 3 3 -manifolds which are tessellated by copies of the regular ideal hyperbolic tetrahedron are geodesically embedded in a complete, finite volume, hyperbolic 4 4 -manifold. This allows us to prove that the complement of the figure-eight knot geometrically bounds a complete, finite volume hyperbolic 4 4 -manifold. This is the first example of geometrically bounding hyperbolic knot complements and, amongst known examples of geometrically bounding manifolds, the one with the smallest volume.