Some computer-assisted topological models of Hilbert fundamental domains

The Hubert modular group ${\text{H}}$ for the integral domain ${\text{O}}({k^{1/2}})$ has a fourdimensional fundamental domain ${\text{R}}$ which should be represented geometrically (like the classic modular group). Computer assistance (by the Argonne CDC 3600) was used for outlining cross sections of the three-dimensional "floor" of ${\text{R}}$, which is a mosaic of an intractably large number of boundary pieces identified under ${\text{H}}$. The cross sections shown here might well contain enough information when $k = 2,3,5,6$ to form some "incidence matrices" and see ${\text{R}}$ (at least) combinatorially. For special symmetrized subgroups of ${\text{H}}$, it is plausible to see homologously independent $2$-spheres in (the corresponding) ${\text{R}}$. The program is a continuation of one outlined in two earlier issues of this journal v. 19, 1965, pp. 594-605, MR 33 #4016, and v. 21, 1967, pp. 76-86, MR 36 #5081.