Axis of the product of two loxodromic isometries Suppose that $X$ and $Y$ are two loxodromic isometries of the hyperbolic space and that the product $XY$ is also a loxodromic element.
We consider the axes of these three elements. I'd like to know if we can say something about the mutual position of these axes, for instance if the fixed points of $XY$ have to respect some special placement with respect to the ones of $X$ and $Y$, or if we can say something about the angles between axes and common perpendicular lines and so on.
I couldn't find any book or paper covering this topic, but also a reference would be greatly appreciated. Thank you.
 A: In dimension $2$ and $3$, you can entirely determine the product geometrically (meaning by that translation length, axis, and angle of rotation around the axis) by drawing a right-angled hexagon. 
Say that you are given $A$ and $B$ two isometries. Draw their common perpendicular $L$, and call $H_L$ the half-turn around $L$. Now, there exist a unique half-turn $H_A$ such that $A$ can be written as $A = H_AH_L$. 
We can do the same thing with $B = H_LH_B$. Thus, you find $AB = H_AH_B$.
The key observation here is that if an isometry $u$ is written as the product of two half-turns, then these half-turns have axis orthogonal to that of $u$, their intersection point with the axis of $u$ are half the translation length of $u$ apart, and they make an angle that is half the rotation angle of $u$ (you can see all this with some "geometric intuition" and a good picture).
Thus, drawing the axis of $H_A$ and $H_B$, you see that they are orthogonal to the axis of $A$ and $B$ respectively, and that their common perpendicular is the axis of $AB$.
A good reference for this is Fenchel's book Elementary Geometry in Hyperbolic Space chapter V.1. It will allow you to prove easily my first claim (that there exist a half-turn $H_A$ such that $A = H_LH_A$ because $L$ is perpendicular to the axis of $A$).
Note that once you've shown this, you don't need to talk about half-turns to compute everything geometrically: simply draw the common perpendicular, then the orthogonal line translated by half the translation length and rotated by half the rotation, do the same for $B$, and draw the common perpendicular : it is the axis of $AB$, and all the remaining information about $AB$ can be read in this right-angled hexagon as well.
If you want some tools that allow you to compute analytically those information, you can also look at Chapter VI of Fenchel's book, that gives many trigonometric formulas for right angled hexagons.
