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Recently I have been going through the book Hyperbolic Knot Theory by Jessica Purcell. In this book, there is an exercise in chapter 5, section 5.6, and exercise number 5.4 (Page - 101). This exercise gives us a presentation of the fundamental group of figure-eight knot complements in $S^3$ the presentation is as follows

$\pi_1(S^3-K)= \langle a,b : yay^{-1}=b \rangle$ where $y=a^{-1}bab^{-1}$ ($K$ the figure-eight knot) and also the representation in $PSL(2,\mathbb{C})$ is given by the matrices

$ A =\begin{bmatrix} 1 & 1\\ 0 & 1 \end{bmatrix} $

$ B=\begin{bmatrix} 1 & 0\\ -\omega & 1 \end{bmatrix} $

where $\omega ^3=1$ and $\Gamma = \langle A,B \rangle$, and $\Gamma$ act on $\mathbb{H}^3$ by isometry (here $\mathbb{H}^3$ is the upper half space model). Now my question is the following:

Are there two elements in $\Gamma$ such that one of them is hyperbolic, say $\alpha$ ( $trace^2$ is real and $>4$), and another on loxodromic, say $\beta$ ($trace^2$ not in the interval $[0, \infty)$), but not hyperbolic such that fixed points of $\alpha$ and $\beta$ are in the same line and the geodesics passing through the fixed points do not intersect?

(More elaborately I can say geodesics $g_{\alpha}$ passing through the fixed points of $\alpha$ and $g_{\beta}$ the geodesic passing through the fixed points of $\beta$, $g_{\alpha}$ and $g_{\beta}$ lie in the same plane but they do not intersect.) NB="lie in the same plane" I mean that it is a hyperbolic plane in upper half-space model

Thanks in advance

Recently I have been going through the book Hyperbolic Knot Theory by Jessica Purcell. Exercise 5.4 (on page 101) gives us a presentation of the fundamental group of $S^3 - K$ where $K$ is the figure-eight knot. The presentation is

$$\pi_1(S^3-K)= \langle a,b : yay^{-1}=b \rangle$$

Here $y=a^{-1}bab^{-1}$. Also we are given a representation into $PSL(2,\mathbb{C})$ by

$$ A =\begin{bmatrix} 1 & 1\\ 0 & 1 \end{bmatrix} \quad \mbox{and} \quad B=\begin{bmatrix} 1 & 0\\ -\omega & 1 \end{bmatrix} $$

Hhere $\omega$ is a non-trivial third root of unity. Taking $\Gamma = \langle A,B \rangle$ we find that $\Gamma$ acts on hyperbolic space $\mathbb{H}^3$ (thinking of $\mathbb{H}^3$ as the upper half space model). Now my question is the following:

Are there two elements $\alpha$ and $\beta$ in $\Gamma$ such that $\alpha$ is hyperbolic (the trace squared is real and greater than four), such that $\beta$ is strictly loxodromic (the trace squared is not in the interval $[0, \infty)$), and such that the fixed points of $\alpha$ and $\beta$ are in the same line and the geodesics passing through the fixed points do not intersect?

(Said another way: Let $g_{\alpha}$ and $g_\beta$ be the axes of $\alpha$ and $\beta$, respectively. So above we want $g_{\alpha}$ and $g_{\beta}$ to lie in a single hyperbolic plane, but not intersect.)

Recently I have been going through the book Hyperbolic Knot Theory by Jessica Purcell. In this book, there is an exercise in chapter 5, section 5.6, and exercise number 5.4 (Page - 101). This exercise gives us a presentation of the fundamental group of figure-eight knot complements in $S^3$ the presentation is as follows

$\pi_1(S^3-K)= <a,b : yay^{-1}=b>$ where $y=a^{-1}bab^{-1}$ ($K= $fig-eight knot) and also the representation in $PSL(2,\mathbb{C})$ is given by the metrices

$ A =\begin{bmatrix} 1 & 1\\ 0 & 1 \end{bmatrix} $

$ B=\begin{bmatrix} 1 & 0\\ -\omega & 1 \end{bmatrix} $

Where $\omega ^3=1$

  and $\Gamma = <A,B>$, and $\Gamma$ act on $\mathbb{H}^3$ by isometry (Here $\mathbb{H}^3$ upper half space model  ) Now my question is does there exist two elements in $\Gamma$ such that one of them is hyperbolic say $\alpha$ ( $trace^2$ is real and $>4$) and another on loxodromic say $\beta$ ($trace^2$ not in the interval $[0, \infty)$  )but not hyperbolic such that fixed points of $\alpha$ and $\beta$ are in the same line and the geodesics passing through the fixed points are not intersecting each other?  (More elaborately I can say geodesics $g_{\alpha}$ passing through the fixed points of $\alpha$ and $g_{\beta}$ the geodesic passing through the fixed points of $\beta$, $g_{\alpha}$ and $g_{\beta}$ lie in the same plane but they do not intersect.) NB="lie in the same plane" I mean that it is a hyperbolic plane in upper half-space model

Thanks in advance

Recently I have been going through the book Hyperbolic Knot Theory by Jessica Purcell. In this book, there is an exercise in chapter 5, section 5.6, and exercise number 5.4 (Page - 101). This exercise gives us a presentation of the fundamental group of figure-eight knot complements in $S^3$ the presentation is as follows

$\pi_1(S^3-K)= \langle a,b : yay^{-1}=b \rangle$ where $y=a^{-1}bab^{-1}$ ($K$ the figure-eight knot) and also the representation in $PSL(2,\mathbb{C})$ is given by the matrices

$ A =\begin{bmatrix} 1 & 1\\ 0 & 1 \end{bmatrix} $

$ B=\begin{bmatrix} 1 & 0\\ -\omega & 1 \end{bmatrix} $

where $\omega ^3=1$ and $\Gamma = \langle A,B \rangle$, and $\Gamma$ act on $\mathbb{H}^3$ by isometry (here $\mathbb{H}^3$ is the upper half space model). Now my question is the following:

Are there two elements in $\Gamma$ such that one of them is hyperbolic, say $\alpha$ ( $trace^2$ is real and $>4$), and another on loxodromic, say $\beta$ ($trace^2$ not in the interval $[0, \infty)$), but not hyperbolic such that fixed points of $\alpha$ and $\beta$ are in the same line and the geodesics passing through the fixed points do not intersect? 

(More elaborately I can say geodesics $g_{\alpha}$ passing through the fixed points of $\alpha$ and $g_{\beta}$ the geodesic passing through the fixed points of $\beta$, $g_{\alpha}$ and $g_{\beta}$ lie in the same plane but they do not intersect.) NB="lie in the same plane" I mean that it is a hyperbolic plane in upper half-space model

Thanks in advance