Hello i am studying triangles in hyperbolic space, and since the distance of points in hyperbolic space is given by the natural logarithm of a cross-ratio, if by a möbius transformation i can represent any motion of the points of this triangle, by showing that a cross-ratio is invariant by möbius transformations i can make an in depth study of equilateral triangles in hyperbolic space.
I have read in a article that some translations, rotations, inversions and homotheties can be represented by möebius transformations but it wasn't clear if any translations, rotations, inversions and homotheties can be represented by a möbius transformation.
If it can be, then can someone please give me some reference in wich this is demonstrated ?
Translations (along a specific axis!) and rotations are Möbius transformations. This includes limit rotations that transport points along horocyclic paths, as a limiting case between translations and rotations.
Inversions (reflections in a hyperbolic line) are anti-Möbius transformations, i.e. combinations of a complex conjugation and a Möbius transformation which preserve real-valued cross ratios, too. Glide-reflections, the combination of a reflection and a translation, are anti-Möbius transformations as well.
Taken together, Möbius transformations and anti-Möbius transformations together (which I would collectively call generalized Möbius transformations, but that term might not be understood by all audiences) can be used to describe all isometries of the hyperbolic plane in one of the Poincaré models.
Homotheties with a scale factor other than $\pm1$ are not possible in hyperbolic geometry, as you cannot change lengths without changing angles. For a scale of $1$ you have the identity, and for $-1$ this is equivalent to a $180°$ rotation so the question about rotation covers this already.
Section 2.2. of my dissertation discusses some of these aspects in more detail, including a closer look at the formulas of the Möbius transformations for the Poincaré disk model.