Symmetries of tilings of Lorentz spaces

We study tilings of the $3$-dimensional simply connected Lorentz manifold of constant curvature. This manifold is modelled on the Lie group
$$G=\widetilde{\operatorname{SU}(1,1)}\cong\widetilde{\operatorname{SL}(2,\mathbf{R})},$$
equipped with the Killing form. The tilings are produced by the fundamental domain construction introduced by the second author. The construction gives Lorentz polyhedra as fundamental domains for the action by left multiplication of a discrete co-compact subgroup of finite level. We determine the symmetry groups of these tilings and discuss the connection with the Seifert fibration of the quotient space. We then give an explicit description of the symmetry group of the tiling in the case when the discrete subgroup is a lift of a triangle group.