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


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.

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


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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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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

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

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

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


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

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

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

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 \(n \ge 5\),

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


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

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

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


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

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)

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Erlandsson, V., Zakeri, S. On Margulis cusps of hyperbolic \(4\)-manifolds. Geom Dedicata 174, 75–103 (2015).

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

Mathematics Subject Classification

  • 22E40
  • 30F40
  • 32Q45