Geometriae Dedicata

, Volume 126, Issue 1, pp 155–175 | Cite as

Fundamental domains in Lorentzian geometry

Original Paper


We consider a discrete subgroup Γ of the simply connected Lie group \(\widetilde{\operatorname{SU}}(1,1)\) of finite level, i.e. the subgroup intersects the centre of \(\widetilde{\operatorname{SU}}(1,1)\) in a subgroup of finite index, this index is called the level of the group. The Killing form induces a Lorentzian metric of constant curvature on the Lie group \(\widetilde{\operatorname{SU}}(1,1)\). The discrete subgroup Γ acts on \(\widetilde{\operatorname{SU}}(1,1)\) by left translations. We describe the Lorentz space form \(\widetilde{\operatorname{SU}}(1,1)/\Gamma\) by constructing a fundamental domain F for Γ. We want F to be a polyhedron with totally geodesic faces. We construct such F for all Γ satisfying the following condition: The image \(\bar\Gamma\) of Γ in PSU(1,1) has a fixed point u in the unit disk of order larger than the index of Γ. The construction depends on the group Γ and on the orbit Γ(u) of the fixed point u.


Lorentz space form Polyhedral fundamental domain Quasihomogeneous singularity Arnold singularity series 

Mathematics Subject Classifications (2000)

53C50 14J17 32S25 51M20 52B10 


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© Springer Science + Business Media B.V. 2007

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität BonnBonnGermany

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