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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= $fig-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 $\mathbb{H}^3$ 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

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  • $\begingroup$ are you sure your B is in PSL2(C) ? $\endgroup$
    – mercio
    May 20 at 9:59
  • $\begingroup$ I am extremely sorry its a typo $\endgroup$
    – T ghosh
    May 20 at 10:21
  • $\begingroup$ The matrix $ B= \begin{bmatrix} 1 & 0\\ -\omega & 1 \end{bmatrix} $ $\endgroup$
    – T ghosh
    May 20 at 10:23
  • $\begingroup$ @mercio Thanks for pointing it out. $\endgroup$
    – T ghosh
    May 20 at 10:24

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