Skip to main content

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

Abstract

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.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Notes

  1. We don’t know if strike triples can actually occur.

References

  1. Apanasov, B.: Cusp ends of hyperbolic manifolds. Ann. Glob. Anal. Geom. 3, 1–11 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  2. Apanasov, B.: Conformal Geometry of Discrete Groups and Manifolds. Walter de Gruyter, Germany (2000)

    Book  MATH  Google Scholar 

  3. Benedetti, R., Petronio, C.: Lectures on Hyperbolic Geometry. Springer, Berlin (2003)

    Google Scholar 

  4. Erlandsson, V.: The Margulis region and screw parabolic elements of bounded type. arXiv:1209.5680. To appear in Bull. Lond. Math. Soc.

  5. Erlandsson, V., Zakeri, S.: A discreteness criterion for groups containing parabolic isometries. arXiv:1304.2298. To appear in Groups, Geometry and Dynamics, Contemp. Math., AMS

  6. Hardy, G., Wright, E.: An Introduction to the Theory of Numbers, 5th edn. Oxford University Press, Oxford (1980)

    Google Scholar 

  7. Herman, M.: Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations. Publ. Math. de l’IHÉS 49, 5–233 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  8. Jørgensen, T.: On discrete groups of Möbius transformations. Am. J. Math. 98, 739–749 (1976)

    Article  Google Scholar 

  9. Kellerhals, R.: Collars in \(\text{ PSL }(2,\mathbb{H})\). Ann. Acad. Sci. Fenn. 26, 51–72 (2001)

    MATH  MathSciNet  Google Scholar 

  10. Kim, Y.: Quasiconformal stability for isometry groups in hyperbolic 4-space. Bull. Lond. Math. Soc. 43, 175–187 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  11. Ohtake, H.: On discontinuous subgroups with parabolic transformations of the Möbius groups. J. Math. Kyoto Univ. 25, 807–816 (1985)

    MATH  MathSciNet  Google Scholar 

  12. Ratcliffe, J.: Foundations of Hyperbolic Manifolds. Springer, Berlin (1994)

    Book  MATH  Google Scholar 

  13. Shimizu, H.: On discontinuous groups operating on the product of the upper half planes. Ann. Math. 77, 33–71 (1963)

    Article  MATH  Google Scholar 

  14. Susskind, P.: The Margulis region and continued fractions, complex manifolds and hyperbolic geometry (Guanajuato, 2001). Contemp. Math. 311, 335–343 (2002)

    Article  MathSciNet  Google Scholar 

  15. Thurston, W.: Three-Dimensional Geometry and Topology, vol. 1. Princeton University Press, Princeton (1997)

    MATH  Google Scholar 

  16. Tukia, P.: Quasiconformal extension of quasisymmetric mappings compatible with a Möbius group. Acta Math. 154, 153193 (1985)

    Article  MathSciNet  Google Scholar 

  17. Waterman, P.: Möbius transformations in several dimensions. Adv. Math. 101, 87–113 (1993)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

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.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Viveka Erlandsson.

Appendix: On arithmetical characterization of presence

Appendix: On arithmetical characterization of presence

The problem we investigate here is when a given denominator \(q_n\) in the continued fraction expansion of an irrational number \(\alpha \) is present in the boundary function \(\fancyscript{B}_{\alpha }\) (see Sect. 3). We need only consider the case where \(a_{n+1}=1\) since Corollary 3.8 guarantees that \(q_n\) is present when \(a_{n+1} \ge 2\). Assuming \(a_{n+1}=1\), the same corollary and the definition of fair triples show that

$$\begin{aligned} q_{n} \ \text {is present} \quad \Longleftrightarrow \quad r_{n-1,n}<r_{n,n+1}. \end{aligned}$$

By the formula (13), this condition can be written as

$$\begin{aligned} q_{n} \ \text {is present} \quad \Longleftrightarrow \quad \frac{\delta _{n}-\delta _{n+1}}{\delta _{n-1}-\delta _{n}} < \frac{q_{n+1}^2-q_{n}^2}{q_{n}^2-q_{n-1}^2}. \end{aligned}$$
(24)

The right side of the inequality in (24) is easily computed:

$$\begin{aligned} \frac{q_{n+1}^2-q_{n}^2}{q_{n}^2-q_{n-1}^2} = \frac{(q_{n}+q_{n-1})^2-q_{n}^2}{q_{n}^2-q_{n-1}^2} = \frac{2q_{n} q_{n-1} + q_{n-1}^2 }{q_{n}^2-q_{n-1}^2} = \frac{2 \mu +1}{\mu ^2-1}, \end{aligned}$$

where

$$\begin{aligned} a_{n} < \mu = \frac{q_{n}}{q_{n-1}} < a_{n}+1. \end{aligned}$$
(25)

To estimate the left side of the inequality in (24), we use the inequalities

$$\begin{aligned} 0.95 x^2 \le \sin ^2 x \le x^2 \quad \text {for} \ |x| \le \frac{\pi }{12} \end{aligned}$$

which can be easily proved using calculus. Since the denominator \(q_6\) is always \(\ge 13\), by Lemma 2.7(ii),

$$\begin{aligned} \pi \Vert q_n \alpha \Vert < \frac{\pi }{q_{n+1}} \le \frac{\pi }{12} \quad (n \ge 5). \end{aligned}$$

It follows that

$$\begin{aligned} 3.8 \pi ^2 \Vert q_n \alpha \Vert ^2 \le \delta _n = 4 \sin ^2 (\pi \Vert q_n \alpha \Vert ) \le 4 \pi ^2 \Vert q_n \alpha \Vert ^2 \quad (n \ge 5). \end{aligned}$$

Introduce the quantity

$$\begin{aligned} \lambda = \frac{\Vert q_{n} \alpha \Vert }{\Vert q_{n+1} \alpha \Vert } \end{aligned}$$

which by Lemma 2.7(iii) satisfies

$$\begin{aligned} a_{n+2} < \lambda < a_{n+2}+1. \end{aligned}$$
(26)

Note that since \(a_{n+1}=1\), we have \(\Vert q_{n-1} \alpha \Vert = \Vert q_{n} \alpha \Vert + \Vert q_{n+1} \alpha \Vert \), which shows

$$\begin{aligned} \frac{\Vert q_{n-1} \alpha \Vert }{\Vert q_{n} \alpha \Vert } = 1 + \lambda ^{-1}. \end{aligned}$$

Thus, for \(n \ge 5\),

$$\begin{aligned} \frac{\delta _{n}-\delta _{n+1}}{\delta _{n-1}-\delta _{n}}&< \frac{4 \Vert q_{n} \alpha \Vert ^2 - 3.8 \Vert q_{n+1} \alpha \Vert ^2}{3.8 \Vert q_{n-1} \alpha \Vert ^2 - 4 \Vert q_{n} \alpha \Vert ^2} \\&= \frac{1-0.95 \lambda ^{-2}}{0.95(1+\lambda ^{-1})^2-1} = \frac{\lambda ^2-0.95}{-0.05 \lambda ^2+1.9 \lambda + 0.95}. \end{aligned}$$

and

$$\begin{aligned} \frac{\delta _{n}-\delta _{n+1}}{\delta _{n-1}-\delta _{n}}&> \frac{3.8 \Vert q_{n} \alpha \Vert ^2 - 4 \Vert q_{n+1} \alpha \Vert ^2}{4 \Vert q_{n-1} \alpha \Vert ^2 - 3.8 \Vert q_{n} \alpha \Vert ^2} \\&= \frac{0.95 - \lambda ^{-2}}{(1+\lambda ^{-1})^2-0.95} = \frac{0.95 \lambda ^2-1}{0.05 \lambda ^2+2 \lambda + 1}. \end{aligned}$$

Introducing the rational functions

$$\begin{aligned} X(t)&=\frac{2t +1}{t^2-1} \\ Y(t)&=\frac{t^2-0.95}{-0.05 t^2+1.9 t + 0.95} \\ Z(t)&=\frac{0.95 t^2-1}{0.05 t^2+2 t + 1}, \end{aligned}$$

the condition (24) and the above estimates can be summarized as

$$\begin{aligned} q_{n} \ \text {is present} \quad \Longleftrightarrow \quad \frac{\delta _{n}-\delta _{n+1}}{\delta _{n-1}-\delta _{n}} < X(\mu ) \end{aligned}$$
(27)

and

$$\begin{aligned} Z(\lambda ) < \frac{\delta _{n}-\delta _{n+1}}{\delta _{n-1}-\delta _{n}} < Y(\lambda ) \quad (n \ge 5), \end{aligned}$$
(28)

where \(\mu , \lambda \) satisfy (25) and (26).

The following can be deduced from (27) and (28) (compare the graphs of \(X,Y,Z\) in Fig. 5).

  • If \(a_{n} \ge 3\) and \(a_{n+2} \ge 3\), then \(\mu , \lambda >3\) and

    $$\begin{aligned} \frac{\delta _{n}-\delta _{n+1}}{\delta _{n-1}-\delta _{n}} > Z(\lambda )>Z(3)>X(3)>X(\mu ). \end{aligned}$$

    so \(q_{n}\) is absent.

  • If \(a_{n}=2\) and \(a_{n+2} \ge 5\), then \(2<\mu <3, \lambda >5\) and

    $$\begin{aligned} \frac{\delta _{n}-\delta _{n+1}}{\delta _{n-1}-\delta _{n}} > Z(\lambda )>Z(5)>X(2)>X(\mu ), \end{aligned}$$

    so \(q_{n}\) is absent.

  • If \(a_{n} \ge 5\) and \(a_{n+2}= 2\), then \(\mu >5, 2<\lambda <3\) and

    $$\begin{aligned} \frac{\delta _{n}-\delta _{n+1}}{\delta _{n-1}-\delta _{n}} > Z(\lambda )>Z(2)>X(5)>X(\mu ), \end{aligned}$$

    so \(q_{n}\) is absent.

  • If \(a_{n}=1\) and \(a_{n+2} \le 2\), then \(1<\mu <2, 1<\lambda <3\) and

    $$\begin{aligned} \frac{\delta _{n}-\delta _{n+1}}{\delta _{n-1}-\delta _{n}} < Y(\lambda )<Y(3)<X(2)<X(\mu ), \end{aligned}$$

    so \(q_{n}\) is present.

  • Finally, if \(a_{n}=2\) and \(a_{n+2}=1\), then \(2<\mu <3, 1<\lambda <2\) and

    $$\begin{aligned} \frac{\delta _{n}-\delta _{n+1}}{\delta _{n-1}-\delta _{n}} < Y(\lambda )<Y(2)<X(3)<X(\mu ), \end{aligned}$$

    so \(q_{n}\) is present.

Fig. 5
figure 5

Graphs of the rational functions \(X, Y\), and \(Z\). Note that \(Y\) has a singularity at \(t \approx 38.5\) (not shown here) but that does not interfere with our estimates on the interval \([1,3]\)

These findings are summarized in Fig. 6. In all other cases, the presence or absence of \(q_n\) also depends on other partial quotients such as \(a_{n-1}, a_{n+3}\), etc.

Fig. 6
figure 6

The locus of presence (blue) and absence (red) of \(q_n\) in the \((a_{n},a_{n+2})\)-plane when \(a_{n+1}=1\). Here we assume \(n \ge 5\). The white cells can go either blue or red depending on other partial quotients. (Color figure online)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Erlandsson, V., Zakeri, S. On Margulis cusps of hyperbolic \(4\)-manifolds. Geom Dedicata 174, 75–103 (2015). https://doi.org/10.1007/s10711-014-0005-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10711-014-0005-0

Keywords

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

Mathematics Subject Classification

  • 22E40
  • 30F40
  • 32Q45