The groups of motions of some three-dimensional maximal mobility geometries (Q721375)

There are sixteen families of three-dimensional maximal mobility geometries. For eight of them, the authors obtain a description in terms of a quadratic ring extension \(A\) of the real numbers. The group of motions is isomorphic to the group \(\mathrm{SL}_2(A)\) and the point stabilizer is a three-dimensional subgroup which is contained in the point stabilizer of the standard action of \(\mathrm{SL}_2(A)\) on the projective line over \(A\).