There are investigated supergroups of some hyperbolic space groups with simplicial fundamental domain. Six simplices considered here from [9] are collected in families F9 (T23, T64), F10 (T21, T49, T61), F29 (T34). All of them have the same symmetry by half-turn h, with axis through the midpoints of edges A0A1 and A2A3. Since that isometry identifies pairs of points, if a supergroup with such smaller fundamental domain exists, it is of index 2. At the side pairings of T34 this half-turn implies additional reflections, equal parameters 2a = 6b, and leads to Family 2, considered in [9]. Other possibility to find supergroups is when the simplices have vertices out of the absolute. In that case we can truncate them by polar planes of the vertices and the new polyhedra are fundamental domains of richer groups.
[1]
A. Prékopa,et al.
Non-Euclidean Geometries: János Bolyai Memorial Volume
,
2014
.
[2]
Classification of Tile-Transitive 3-Simplex Tilings and Their Realizations in Homogeneous Spaces
,
2006
.
[3]
Emil Molnár,et al.
Classification of Solid Transitive Simplex Tilings in Simply Connected 3-Spaces II. Metric realizations of the maximal simplex tilings
,
1997
.
[4]
Polyhedron complexes with simply transitive group actions and their realizations
,
1992
.
[5]
B. Maskit,et al.
On Poincaré's theorem for fundamental polygons
,
1971
.
[6]
Emil Molnár,et al.
The projective interpretation of the eight 3-dimensional homogeneous geometries.
,
1997
.