# Moscow Mathematical Journal

Volume 20, Issue 2, April–June 2020  pp. 257–276.

Symmetries of Tilings of Lorentz Spaces

Authors:  Nasser Bin Turki (1) and Anna Pratoussevitch (2)
Author institution:(1) Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
(2) Department of Mathematical Sciences, University of Liverpool, Peach Street, Liverpool L69 7ZL, United Kingdom

Summary:

We study tilings of the 3-dimensional simply connected Lorentz manifold of constant curvature. This manifold is modelled on the Lie group $$G=\widetilde{\operatorname{SU}(1,1)}\cong\widetilde{\operatorname{SL}(2,\mathbb{R})},$$ 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.

2010 Math. Subj. Class. Primary: 52C22, 53C50; Secondary: 51M20, 52B10, 32S25.

Keywords: Tilings, symmetries of tilings, Lorentz space forms, polyhedral fundamental domain, quasi-homogeneous singularity.