Picard horn

A Picard horn, also called the Picard topology or Picard model, is one of the oldest known hyperbolic 3-manifolds, first described by Émile Picard in 1884. The manifold is the quotient of the upper half-plane model of hyperbolic 3-space by the projective special linear group, ${\displaystyle \operatorname {PSL} _{2}(\mathbf {Z} [i])}$. It was proposed as a model for the shape of the Universe in 2004. The term "horn" is due to pseudosphere models of hyperbolic space.

A modern description, in terms of fundamental domain and identifications, can be found in section 3.2, page 63 of Grunewald and Huntebrinker, along with the first 80 eigenvalues of the Laplacian, tabulated on page 72, where ${\displaystyle \Upsilon _{1}}$ is a fundamental domain of the Picard space.

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