I am reading a paper and they mention the following: "Two isometries of the hyperbolic plane are said to be co-parallel if they have disjoint axes and the motions along these axes induce the opposite orientations on the region they bound"
I am very confused by what is meant by that (visually w.r.t to the fixed points) in the upper-half plane. Consider two hyperbolic matrices $A,B$ and assume they have disjoint axis. Denote $a^+,a^-$ ($b^+,b^-$) as attracting/repelling fixed points of $A$ (of $B$) respectively. In my head, what the above means is that either $b^+>b^-$ and $a^+>a^-$ or $b^+<b^-$ and $a^+<a^-$. Visually, the translation axis "point" in the same direction when pointing towards the attracting point. However, doing some computations using the results in the paper, my understanding is wrong. Could anyone explain what is meant by the above?
For example: $$A=\left( \begin{array}{cc} -1.01179 & -2.92967 \\ 2.38967 & 5.93101 \\ \end{array} \right),B=\left( \begin{array}{cc} -0.244863 & 2.47968 \\ -0.635634 & 2.35302 \\ \end{array} \right)$$ Are co-parallel, but $a^+=-0.51231,a^-= -2.39302$ and $b^+=1.5192,b^-= 2.56787$
Thank you!
Suppose that you have two oriented arcs on the boundary of a domain $D$ in the plane. Then the two arcs define the same orientation on if and only if, as you travel along these arcs in the direction of their orientation, the domain is to the same side from you. (I can add a formal definition if you wish.) Now, consider two examples. In the first example the two arcs define the same orientation on $D$: As you travel along the both arcs, the domain is to the left from you.
In the second example, the arcs define opposite orientations on $D$: As you travel along the red arc, the domain is to the left from you, while as you travel along the blue arc, the domain is to the right from you.
Note that in both cases, the inequalities $a_-< a_+, b_-< b_+$ are satisfied. Hence, your conjecture about orientation in terms of stated inequalities between $a_\pm, b_\pm$ is (in general) false. This probably explains why you are getting answers different from the ones in the paper you are reading.
More precisely, your conjecture fails if the geodesic arcs bound disjoint half-disks in the upper half-plane. The conjecture holds if the arcs bound nested half-disks.