On the arithmetic 4-orbifolds associated to integral quaternary quadratic forms of index 2

Groups acting properly and discontinuously on the Cartesian product (Formula presented.) of two hyperbolic planes are termed hyperabelian by Picard. The automorphism group (Formula presented.) of a quaternary integral quadratic form f of index 2 is an example of a hyperabelian group. Hence the quotient orbifold (Formula presented.) of the action of (Formula presented.) on (Formula presented.) is a 4-dimensional arithmetic orbifold, endowed with a natural (Formula presented.)-geometry. Plücker coordinates are used to understand (Formula presented.). A real automorphism U of (Formula presented.) induces a real automorphism (Formula presented.) of (Formula presented.) in such a way that if (Formula presented.) then (Formula presented.) is an automorph of the Klein quadratic form k. It is proved that the converse is true. That is, given an automorph (Formula presented.) of k there is (Formula presented.) such that (Formula presented.), so that the proper automorphism group of the Klein quadric is isomorphic to (Formula presented.) via (Formula presented.). This is used to obtain the automorphism group of the quadratic line complex of line tangents to a quadric in projective space (Formula presented.). With this, a description is given of the automorphism group of a quaternary integral quadratic form of index 2