Geometriae Dedicata

, Volume 174, Issue 1, pp 75–103 | Cite as

On Margulis cusps of hyperbolic \(4\)-manifolds

Original Paper


We study the geometry of the Margulis region associated with an irrational screw translation \(g\) acting on the \(4\)-dimensional real hyperbolic space. This is an invariant domain with the parabolic fixed point of \(g\) on its boundary which plays the role of an invariant horoball for a translation in dimensions \({\le }\)3. The boundary of the Margulis region is described in terms of a function \(\fancyscript{B}_{\alpha }: [0,\infty ) \rightarrow {\mathbb {R}}\) which solely depends on the rotation angle \(\alpha \in {\mathbb {R}}/{\mathbb {Z}}\) of \(g\). We obtain an asymptotically universal upper bound for \(\fancyscript{B}_{\alpha }(r)\) as \(r \rightarrow \infty \) for arbitrary irrational \(\alpha \), as well as lower bounds when \(\alpha \) is Diophantine and the optimal bound when \(\alpha \) is of bounded type. We investigate the implications of these results for the geometry of Margulis cusps of hyperbolic \(4\)-manifolds that correspond to irrational screw translations acting on the universal cover. Among other things, we prove bi-Lipschitz rigidity of these cusps.


Screw translation Hyperbolic \(4\)-space Horoball  Cusp Margulis region Continued fractions 

Mathematics Subject Classification

22E40 30F40 32Q45 



We are grateful to Ara Basmajian for sharing his knowledge and lending his support at various stages of this project. We also thank Perry Susskind for useful conversations on the topics discussed here.


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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Aalto Science InstituteAalto UniversityEspooFinland
  2. 2.Department of MathematicsQueens College and Graduate Center of CUNYNew YorkUSA

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