The theory of finite groups generated by reflections in the real $ n $-dimensional space is a 150 years old rich area. Among these the building blocks are the Weyl groups. All reflection subgroups of these have recently been described in full. We are interested in the proprties of the (infinite) permutation actions on the cosets of these subgroups.

For a TDK submission we expect the solution of some special cases.

For a PhD thesis we expect the solution and description of all cases.